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Algebra

Algebra Practice Test: Practice Test 2

Practice Test 2 for Algebra: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1 of 25

An investment formula is given by $A = P(1 + r)^t + 500(1 + r)^{t-2}$ . To better understand the structure of this expression, which decomposition most clearly reveals the relationship between the terms?

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Question 1

$A = (1 + r)^t[P + \frac{500}{(1 + r)^2}]$ , showing a common factor of $(1 + r)^t$ with a bracketed expression
$A = P(1 + r)^t + \frac{500}{(1 + r)^2}(1 + r)^t$ , showing both terms as multiples of $(1 + r)^t$ with different coefficients
$A = (P + 500)(1 + r)^t - 500(1 + r)^2$ , showing the sum distributed across exponential terms
$A = (1 + r)^{t-2}[P(1 + r)^2 + 500]$ , showing a common exponential factor and a bracketed sum depending on $P$ (correct answer)

Explanation: When you encounter algebraic expressions with multiple exponential terms, look for opportunities to factor out common elements to reveal the underlying structure. This process helps simplify complex expressions and makes relationships between terms clearer. To find the best decomposition, you need to identify the greatest common factor among the exponential terms. In $A = P(1 + r)^t + 500(1 + r)^{t-2}$ , both terms contain powers of $(1 + r)$ . Since $t-2$ is smaller than $t$ , you can factor out $(1 + r)^{t-2}$ from both terms. From the first term: $P(1 + r)^t = P(1 + r)^{t-2} \cdot (1 + r)^2$ From the second term: $500(1 + r)^{t-2}$ remains unchanged. Factoring gives: $A = (1 + r)^{t-2}[P(1 + r)^2 + 500]$ This matches answer choice D, which correctly shows $(1 + r)^{t-2}$ as the common exponential factor with a bracketed expression containing the remaining terms. Answer A incorrectly factors out $(1 + r)^t$ , which would require dividing the second term by $(1 + r)^2$ , but this creates an incorrect negative exponent situation. Answer B rewrites the expression without actually factoring, just rearranging terms. Answer C attempts to distribute incorrectly and changes the mathematical meaning entirely. Study tip: When factoring exponential expressions, always factor out the term with the smallest exponent first. This ensures you're pulling out the true greatest common factor and reveals the clearest structural relationships.

Question 2

A store sign lists a product weight as 1.50 lb. The scale used in the store reads to the nearest 0.1 lb. Identify the false precision in the statement.

The unit lb is too precise for a scale
The 1.50 suggests precision to the nearest 0.01 lb, which the scale cannot measure (correct answer)
The 1.50 is too rounded; it should be 1.5 exactly
The weight should be reported in ounces only

Explanation: This question tests your understanding that the precision of your reported answer should match the precision of your measurements—you can't claim accuracy beyond what your measurement tools or methods allow. False precision is reporting too many digits: saying 'we need 3.7142857 people for the job' is silly—you can't have fractional people! Or 'the stick is 12.384729 cm' when you measured with a ruler marked in cm (so really 12 cm or maybe 12.4 cm at most). Excessive decimals don't make you look more accurate; they show you don't understand measurement limitations. Round appropriately! The statement '1.50 lb' shows false precision: the scale measures to nearest 0.1 lb, so reporting to hundredths suggests finer precision. With measurement to tenths, we can only reliably report to tenths. The extra zero is unjustified—it suggests accuracy beyond what the measurement method actually provides. Appropriate reporting: 1.5 lb. Choice B correctly identifies that the 1.50 suggests precision to the nearest 0.01 lb, which the scale cannot measure which appropriately reflects measurement capability. Choice C over-rounds, losing precision unnecessarily: the measurements justify reporting to tenths, but this choice suggests something incorrect. While not as bad as false precision, unnecessary rounding loses information. If your measurements support tenths, report to tenths; don't round to ones unless needed! False precision red flags: (1) More decimals than your measuring tool can detect, (2) Fractional counts of discrete things, (3) More sig figs than any input had, (4) Excessive precision for the context (measuring a room 'exactly 12.0000000 feet'). If you see these in your work or others', recognize it as inappropriate precision. Better to round sensibly than claim unjustified accuracy! Trailing zeros matter: 3.5 vs 3.50 vs 3.500 represent different claimed precision (tenths vs hundredths vs thousandths). Only include trailing zeros after the decimal if your measurement actually supports that precision! If measured to hundredths, write 3.50 (showing you measured hundredths). If measured to tenths, write 3.5 (don't add fake zeros). Trailing zeros after decimal are significant—they're precision claims!

Question 3

Given $f(x) = x^2 - 4x - 12$ , a student factors this as $(x - 6)(x + 2)$ and identifies key features. Which analysis of the parabola's symmetry is correct?

The parabola is symmetric about the line $x = 4$ passing through the y-intercept
The parabola is symmetric about the line $x = -2$ passing through one zero
The parabola is symmetric about the line $x = 6$ passing through one zero
The parabola is symmetric about the line $x = 2$ passing through the vertex (correct answer)

Explanation: When analyzing parabolas, the axis of symmetry always passes through the vertex and lies exactly halfway between the zeros (x-intercepts). This vertical line is where the parabola "mirrors" itself. From the factored form $(x - 6)(x + 2)$ , you can find the zeros by setting each factor equal to zero: $x = 6$ and $x = -2$ . The axis of symmetry falls at the midpoint between these zeros: $x = \frac{6 + (-2)}{2} = \frac{4}{2} = 2$ . Since parabolas have their vertex on the axis of symmetry, and the vertex is the minimum point for upward-opening parabolas like this one (positive leading coefficient), the axis of symmetry $x = 2$ does indeed pass through the vertex. Choice A is incorrect because $x = 4$ is not the axis of symmetry, and the y-intercept (found by setting $x = 0$ : $f(0) = -12$ ) is a point on the parabola, not related to the axis of symmetry. Choice B places the axis of symmetry at $x = -2$ , which is actually one of the zeros, not the line halfway between them. Choice C similarly places the axis at $x = 6$ , the other zero, rather than between the zeros. Study tip: Always remember that for any parabola, the axis of symmetry is the average of the two zeros: $x = \frac{\text{zero}_1 + \text{zero}_2}{2}$ . This line passes through the vertex and divides the parabola into two mirror-image halves.

Question 4

Solve the system: $\begin{cases} 3x-y=7 \\ x+y=5 \end{cases}$ What is the solution $(x, y)$ ?

$(3,2)$ (correct answer)
$(2,3)$
$(4,1)$
$(1,4)$

Explanation: This question tests your ability to solve systems of linear equations—finding the $(x, y)$ pair that makes both equations true at the same time. The elimination method (also called addition method) works by adding or subtracting the equations to make one variable disappear: if you have $x + y = 5$ and $x - y = 1$ , adding them gives $2x = 6$ because the y terms cancel out. Then solve for x, and use that to find y! Add the equations: $(3x - y) + (x + y) = 7 + 5$ , which simplifies to $4x = 12$ , so $x = 3$ ; then substitute into $x + y = 5$ to get $3 + y = 5$ , so $y = 2$ . Choice A is correct because it gives the $(x, y)$ pair $(3, 2)$ that satisfies both equations when you substitute back to check: $3(3) - 2 = 9 - 2 = 7$ and $3 + 2 = 5$ . Switching x and y might lead to choice B $(2, 3)$ , but remember to solve step by step. Here's how to choose a method: if one equation is already solved for a variable (like $y = 3x + 1$ ), use substitution—it's set up perfectly! If the coefficients of one variable are opposites (like $2x$ and $-2x$ ) or the same (like $3y$ and $3y$ ), use elimination—one variable will cancel nicely. And you can always graph both lines to see where they cross! With practice, you'll spot the easiest method for each system.

Question 5

Where is the vertex of $g(x)=|x-3|+2$ ?

$(2,3)$
$(3,-2)$
$(3,2)$ (correct answer)
$(-3,2)$

Explanation: This question tests your understanding of how to graph absolute value functions and identify their key features like the vertex. Absolute value functions like f(x) = |x - h| + k create a V-shape with the vertex (point where direction changes) at (h, k). The graph is made of two linear pieces: one with positive slope for x ≥ h, one with negative slope for x < h. You can think of absolute value as 'distance from zero,' which is why |3| = 3 and |-3| = 3—both are 3 units away from zero! For g(x) = |x - 3| + 2, the vertex is at (3, 2)—this is where the inside equals zero. The graph makes a V: for x < 3, the slope is -1 (going down left toward vertex), and for x ≥ 3, the slope is +1 (going up right from vertex). Plot the vertex (3, 2), then draw two straight lines forming a V-shape: left side going down with slope -1, right side going up with slope +1. If there's a coefficient like 2|x - h|, the V is steeper! Choice C correctly locates the vertex at (3,2) because for |x - 3| + 2, the vertex is where x - 3 = 0, so x = 3, and then +2 gives y = 2. Choice A places the vertex at the wrong location: for |x - 3| + 2, the vertex is where the inside equals zero (x - 3 = 0), giving x = 3. So vertex is at (3, 2), not (3, -2)—watch the sign of the constant term! Absolute value vertex trick: the vertex is at (h, k) from f(x) = |x - h| + k, but watch the sign! |x - 3| has vertex at x = 3 (positive), while |x + 3| = |x - (-3)| has vertex at x = -3 (negative). The value that makes the inside equal zero is where the V points! Quick shape recognition: Absolute value = sharp V-shape with vertex at the point. Memorizing these characteristic shapes helps you sketch quickly and recognize function types from graphs!

Question 6

A community garden is planting tomatoes and peppers. Let $x$ = number of tomato plants and $y$ = number of pepper plants.

Constraints:

There is space for at most 30 plants total: $x+y\le 30$ .
Tomato plants must be at least 10: $x\ge 10$ .
Each tomato plant needs 2 minutes/day of watering and each pepper plant needs 3 minutes/day; they have at most 75 minutes/day: $2x+3y\le 75$ .
$x$ and $y$ must be nonnegative whole numbers.

Is the point $(12, 6)$ a viable solution? Explain by identifying whether it satisfies all constraints.

Viable; it satisfies $x+y\le 30$ , $x\ge 10$ , $2x+3y\le 75$ , and uses nonnegative whole numbers. (correct answer)
Nonviable; it violates the space constraint because $12+6>30$ .
Nonviable; it violates the watering constraint because $2(12)+3(6)>75$ .
Nonviable; it violates the minimum tomato constraint because $12<10$ .

Explanation: This question tests your ability to translate real-world constraints into mathematical inequalities (and equations) and determine whether potential solutions are viable—meaning they satisfy all constraints and make sense in the context. A solution is viable (feasible) if it satisfies EVERY SINGLE constraint AND makes sense in the real-world context (like no negative quantities, whole items when you can't buy half an item, etc.). If even one constraint is violated, or if the solution is unrealistic, it's nonviable. Think of constraints like security checkpoints—you need to pass through all of them! To check if (12, 6) is viable, we substitute into each constraint: Space constraint: 12 + 6 = 18 ≤ 30 ✓ (true). Minimum tomatoes: 12 ≥ 10 ✓ (true). Watering time: 2(12) + 3(6) = 24 + 18 = 42 ≤ 75 ✓ (true). Also checking context: 12 and 6 are both nonnegative whole numbers ✓. Conclusion: viable because all constraints are satisfied. Choice A is correct because it properly checks all constraints showing (12, 6) is viable and correctly identifies that all inequalities are satisfied. Choice B says the point is nonviable but doesn't check correctly: it claims 12 + 6 > 30, but 12 + 6 = 18, which is less than 30. To determine viability, you MUST check every single constraint accurately—missing even one or calculating incorrectly can lead to rejecting a perfectly good solution! The viability-checking procedure: Make a checklist of every constraint. For each one, substitute the point and check if it's satisfied. Write 'Yes' or 'No' next to each constraint. If even one 'No' appears, the solution is nonviable—identify which constraint(s) failed. Also do a reality check: negative quantities? fractional items when must be whole? These context violations also make solutions nonviable!

Question 7

A parabola is defined by $h(x)=-(x-4)^2+1$ . What is the maximum value of $h(x)$ ?

$-1$
$4$
$5$
$1$ (correct answer)

Explanation: This question tests your understanding of how to identify key features of quadratic functions and their graphs, specifically the maximum value. For a quadratic function in the form y = a(x - h)² + k, the vertex is the point (h, k), which is the highest point if the parabola opens down (a < 0) or the lowest point if it opens up (a > 0). This quadratic is in vertex form y = a(x - h)² + k, where we can read the vertex directly: h(x) = -(x - 4)² + 1 has vertex (4, 1). Since a = -1 is negative, the parabola opens down, making the vertex the maximum point. The maximum value is the y-coordinate of the vertex, which is 1. Choice C is correct because it properly identifies the maximum value as 1, which is the y-coordinate of the vertex (4, 1). Great job identifying this! Choice B gives 4, which is the x-coordinate of the vertex, not the maximum value. The maximum value is always the y-coordinate of the highest point, not the x-coordinate. We need the height, not the horizontal position! To remember parabola direction: positive a is a 'happy face' parabola opening up with a minimum, negative a is a 'sad face' opening down with a maximum. This visual trick helps you remember which is which!

Question 8

The function $p(x)=\sqrt{x}$ assigns one output to each input in its domain. What is the range of $p$ ? (Answer in interval notation.)

$(-\infty,\infty)$
$[0,\infty)$ (correct answer)
$(-\infty,0]$
$(0,\infty)$

Explanation: This question tests your understanding of what functions are, and how to determine their domains (possible inputs) and ranges (possible outputs). The range is the set of all possible output values (y-values) the function can produce: for a quadratic that opens upward with vertex at (2, -3), the range is all y-values greater than or equal to -3 because the parabola goes up from that minimum point forever. For p(x) = √x, the square root function only produces non-negative outputs—you can never get a negative number from a principal square root. The smallest output is √0 = 0, and as x increases, √x increases without bound. Choice B is correct because [0,∞) represents all non-negative real numbers, with the square bracket showing 0 is included as the minimum possible output. Choice C incorrectly suggests negative outputs are possible, but √x is always non-negative by definition. For finding range: linear functions usually have range = all real numbers. For quadratics, find the vertex first—if it opens up, range is y ≥ (vertex y-value); if it opens down, range is y ≤ (vertex y-value). For square roots, range is usually y ≥ 0. The function type tells you a lot!

Question 9

If $\sqrt[3]{x^2} = x^{\frac{a}{b}}$ where $\frac{a}{b}$ is in lowest terms, what is the value of $a + b$ ?

$8$
$3$
$6$
$5$ (correct answer)

Explanation: When you encounter expressions with radicals and exponents, the key is converting everything to exponential form using the rule that $\sqrt[n]{x^m} = x^{\frac{m}{n}}$ . Let's rewrite the left side of the equation. The cube root $\sqrt[3]{x^2}$ can be expressed as $x^{\frac{2}{3}}$ . This means our equation becomes: $x^{\frac{2}{3}} = x^{\frac{a}{b}}$ Since the bases are the same (both are $x$ ), the exponents must be equal: $\frac{a}{b} = \frac{2}{3}$ We need to check if $\frac{2}{3}$ is already in lowest terms. Since 2 and 3 share no common factors other than 1, this fraction is already reduced. Therefore, $a = 2$ and $b = 3$ , giving us $a + b = 2 + 3 = 5$ . Looking at the wrong answers: Choice A (8) might come from incorrectly adding exponents or misapplying power rules. Choice B (3) represents just the value of $b$ alone, which suggests forgetting to add $a$ . Choice C (6) could result from multiplying $a \times b = 2 \times 3 = 6$ instead of adding them. The answer is D. Study tip: Always convert radicals to fractional exponents first—use $\sqrt[n]{x^m} = x^{\frac{m}{n}}$ . Then remember that when bases are equal, exponents must be equal. Finally, double-check that your fraction is in lowest terms by ensuring the numerator and denominator have no common factors.

Question 10

For the linear function $p(x)=4x-7$ , what is the average rate of change of $p(x)$ from $x=-2$ to $x=3$ ?

$-4$
$4$ (correct answer)
$\frac{4}{5}$
$20$

Explanation: This question tests your understanding of average rate of change, which is a super important concept connecting slope, functions, and real-world rates like speed or growth. For a linear function, the average rate of change is the same as the slope and doesn't depend on which interval you choose—the function changes at a constant rate everywhere. But for nonlinear functions like quadratics, the average rate of change can be different over different intervals. To find the average rate of change of p(x)=4x-7 from x=-2 to x=3, we first evaluate at the endpoints: p(-2)=4*(-2)-7=-15 and p(3)=4*3-7=5. Then we use the formula: average rate = [5 - (-15)]/(3 - (-2)) = 20/5 = 4. That's it! Choice B is correct because it properly calculates [p(3) - p(-2)]/(3 - (-2)) = 20/5 = 4, getting both the arithmetic and the sign right! Choice A flips the order in the numerator or denominator, calculating [p(-2) - p(3)]/(3 - (-2)) = -20/5 = -4, which changes the sign of the answer. The formula is always (later y - earlier y)/(later x - earlier x), maintaining consistent order! Here's a way to remember: average rate of change is just slope between two points. If you can find slope, you can find average rate of change—it's the same calculation! For linear functions, this slope is constant. For curves, the slope of the secant line connecting two points gives you the average rate over that interval.

Question 11

Divide and simplify (write your answer as a simplified rational expression): $\frac{x^2-4}{x+1}\div\frac{x+2}{x+1}$

$\frac{x^2-4}{x+2}$
$\frac{x-2}{x+1}$
$\frac{x+2}{x^2-4}$
$x-2$ (correct answer)

Explanation: This question tests your understanding of how to divide rational expressions—algebraic fractions that work just like regular fractions but with variables. Dividing rational expressions uses the 'keep, change, flip' rule from numeric fractions: to divide by a fraction, multiply by its reciprocal. So $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$ . Flip the second fraction (reciprocal), change ÷ to ×, then multiply! For $\frac{x^2-4}{x+1} \div \frac{x+2}{x+1}$ , flip the second to get $\frac{x^2-4}{x+1} \cdot \frac{x+1}{x+2}$ , cancel the (x+1) terms, factor $x^2-4$ as (x-2)(x+2), then cancel (x+2) to leave $x-2$ . Choice A correctly divides and simplifies to $x-2$ by factoring and canceling common factors after applying the reciprocal. For instance, choice B stops after flipping but doesn't fully simplify—keep going until no common factors remain! The golden rule for multiplying and dividing rationals: factor everything you can BEFORE you multiply or cancel. This prevents working with huge expressions and catches opportunities to simplify.

Question 12

A quadratic equation of the form $x^2 + bx + c = 0$ has solutions $x = 3 \pm 2i$ . What is the value of $b + c$ ?

$19$
$13$
$7$ (correct answer)
$-7$

Explanation: When you see a quadratic equation with complex solutions, you need to work backwards from the solutions to find the original equation. The key insight is that if a quadratic has real coefficients, complex solutions always come in conjugate pairs. Given the solutions $x = 3 \pm 2i$ , you have $x = 3 + 2i$ and $x = 3 - 2i$ . To find the original quadratic, use the fact that if $r$ and $s$ are solutions, then the equation is $(x - r)(x - s) = 0$ . So: $(x - (3 + 2i))(x - (3 - 2i)) = 0$ Expanding this: $((x - 3) - 2i)((x - 3) + 2i) = 0$ This is a difference of squares pattern: $(a - b)(a + b) = a^2 - b^2$ $(x - 3)^2 - (2i)^2 = 0$ $(x - 3)^2 - 4i^2 = 0$ Since $i^2 = -1$ : $(x - 3)^2 + 4 = 0$ $x^2 - 6x + 9 + 4 = 0$ $x^2 - 6x + 13 = 0$ Therefore, $b = -6$ and $c = 13$ , so $b + c = -6 + 13 = 7$ . Looking at the wrong answers: A) $19$ likely comes from adding the coefficients incorrectly. B) $13$ is just the value of $c$ alone, missing the $b$ term. D) $-7$ probably results from sign errors during expansion. Remember: when working backwards from complex solutions, always use the conjugate pair formula and watch your signs carefully during algebraic manipulation.

Question 13

The profit $P$ (in dollars) from selling $x$ items is given by $P(x) = -3x^2 + 24x - 36$ . How many items should be sold to maximize profit, and what is the maximum profit?

4 items; $12 profit (correct answer)
8 items; $60 profit
4 items; $-36 profit
12 items; $108 profit

Explanation: Complete the square: $P(x) = -3x^2 + 24x - 36 = -3(x^2 - 8x) - 36 = -3(x^2 - 8x + 16 - 16) - 36 = -3((x-4)^2 - 16) - 36 = -3(x-4)^2 + 48 - 36 = -3(x-4)^2 + 12$ . Maximum profit is $12 when $x = 4$ . Choice B incorrectly uses $x = 8$ (from $-24/(-6)$ ) and calculates profit incorrectly. Choice C has correct $x$ but uses the constant term as profit. Choice D uses $x = 12$ (from $24/2$ ) and calculates profit incorrectly.

Question 14

If $g(x)=x^2+3x-4$ , what is $g(-2)$ ?

$-6$ (correct answer)
$-14$
$6$
$-2$

Explanation: This question tests your understanding of function notation and how to evaluate functions by substituting input values. Function notation g(x) tells us the rule for calculating outputs from inputs: when you see g(-2), it means 'substitute -2 for every x in the function formula,' like filling in a blank everywhere you see x. When evaluating with a negative number like g(-2), we need to be extra careful with signs! Substituting -2 for x in g(x) = x² + 3x - 4, we get g(-2) = (-2)² + 3(-2) - 4, which equals 4 + (-6) - 4 = 4 - 6 - 4 = -6. Notice how the parentheses around -2 help keep track of the negative! Choice A is correct because it properly substitutes -2 for x in the function and calculates accurately: (-2)² = 4, 3(-2) = -6, so 4 + (-6) - 4 = -6. Nice work if you got this! Choice B gives -14, which results from forgetting that (-2)² = 4 (positive!) and incorrectly calculating it as -4. Remember, a negative number squared is always positive! When substituting negative numbers, always use parentheses to protect yourself from sign errors: write g(-2) = (-2)² + 3(-2) - 4, not g(-2) = -2² + 3(-2) - 4. The parentheses keep everything clear!

Question 15

Let $f(x)=2x$ and $g(x)=x^2+1$ . What is $f(g(3))$ ?

$20$ (correct answer)
$19$
$8$
$2(3^2)+1=19$

Explanation: This question tests your understanding of function composition—taking the output of one function and using it as the input for another function. To evaluate a composition at a specific number like f(g(3)): Step 1 - find g(3) by substituting 3 into g; Step 2 - take that answer and substitute it into f. If g(3) = 10, then you need f(10). It's like a relay race where g passes its output to f! To find f(g(3)), we work inside-out: First, g(3) = 3² + 1 = 9 + 1 = 10. Now we take this result and plug it into f: f(10) = 2 * 10 = 20. So f(g(3)) = 20. Two separate evaluations, one after the other! Choice A correctly evaluates the composition by working inside-out, giving 20. Choice D has the right process but makes an arithmetic error by doing 2(3²) + 1 instead of 2(3² + 1), leading to 19. When composing functions, there are multiple steps where calculation errors can creep in—always double-check each substitution and simplification! For evaluating at a specific number: do it in TWO separate steps: Step 1: Find g(input) = some number. Step 2: Find f(that number) = final answer. Write down the intermediate answer from Step 1 before moving to Step 2—this prevents mistakes and makes checking easier!

Question 16

Two students complete the square for the same equation $x^2 - 8x + 7 = 0$ . Student A writes $(x - 4)^2 = 9$ and Student B writes $(x - 4)^2 - 9 = 0$ . Which statement about their work is correct?

Only Student B's form is correct because all terms should be on one side
Only Student A's form is correct because it matches the required format
Both forms are mathematically equivalent and correct for solving the equation (correct answer)
Both students made errors because the correct form should be $(x + 4)^2 = 9$

Explanation: When you're completing the square, there are multiple valid ways to express your final result. The key is understanding that algebraically equivalent forms are all correct as long as they help you solve the equation. Let's verify both students completed the square correctly. Starting with $x^2 - 8x + 7 = 0$ , you take half of the coefficient of $x$ (which is $-8/2 = -4$ ), square it to get $16$ , then add and subtract this value: $x^2 - 8x + 16 - 16 + 7 = 0$ . This gives you $(x - 4)^2 - 9 = 0$ , which can be rearranged to $(x - 4)^2 = 9$ . Both forms are mathematically sound. Student A's form $(x - 4)^2 = 9$ isolates the perfect square, making it easy to take the square root of both sides. Student B's form $(x - 4)^2 - 9 = 0$ keeps everything on one side, which some prefer. Both lead to the same solutions: $x = 7$ or $x = 1$ . Choice A is wrong because there's no mathematical requirement that all terms stay on one side when completing the square. Choice B is incorrect because there's no single "required format" - both approaches work. Choice D contains a sign error; since we have $-8x$ , we need $(x - 4)^2$ , not $(x + 4)^2$ . The correct answer is C because both forms are mathematically equivalent and will yield the same solutions. Study tip: When completing the square, focus on whether the algebra is correct rather than the specific format. Multiple equivalent forms can all be valid.

Question 17

Evaluate $27^{2/3}$ .

$6$
$12$
$9$ (correct answer)
$18$

Explanation: This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Rational exponents give us an exponential way to write roots: x^(1/n) means the nth root of x (like x^(1/2) = √x and x^(1/3) = ∛x), and more generally, x^(m/n) means take the nth root of x, then raise it to the mth power (or do the power first, then the root—either order works!). To evaluate 27^(2/3), we can think of it as (∛27)² or ∛(27²). Since ∛27 = 3 (because 3³ = 27), we get 3² = 9. Alternatively, 27² = 729, and ∛729 = 9. Choice C is correct because 27^(2/3) = (∛27)² = 3² = 9. Choice A gives 6, which might come from incorrectly calculating 27 × 2/3 = 18, then dividing by 3, but that's not how rational exponents work. For simplifying with exponent properties, convert all radicals to exponential form first if they aren't already: √x becomes x^(1/2), ∛x becomes x^(1/3), etc. Then use your exponent rules (add when multiplying, subtract when dividing, multiply when doing power of a power). Finally, convert back to radical form if that's what's asked for!

Question 18

Rewrite $16^{3/4}$ using radicals, then evaluate.

$12$
$8$ (correct answer)
$4$
$6$

Explanation: This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Rational exponents give us an exponential way to write roots: $x^{1/n}$ means the nth root of x (like $x^{1/2} = \sqrt{x}$ and $x^{1/3} = \sqrt[3]{x}$ ), and more generally, $x^{m/n}$ means take the nth root of x, then raise it to the mth power (or do the power first, then the root—either order works!). To evaluate $16^{3/4}$ , first convert to radical form: $16^{3/4} = \sqrt[4]{16^3}$ or $(\sqrt[4]{16})^3$ . Since $\sqrt[4]{16} = 2$ (because $2^4 = 16$ ), we get $2^3 = 8$ . Choice C is correct because $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$ . Choice B gives 12, which might come from incorrectly calculating 16 × 3/4 = 12, but that's not how rational exponents work. For simplifying with exponent properties, convert all radicals to exponential form first if they aren't already: $\sqrt{x}$ becomes $x^{1/2}$ , $\sqrt[3]{x}$ becomes $x^{1/3}$ , etc. Then use your exponent rules (add when multiplying, subtract when dividing, multiply when doing power of a power). Finally, convert back to radical form if that's what's asked for!

Question 19

Classify each expression as rational or irrational and choose the option that correctly justifies all three:

(1) $\frac{2}{5}+\frac{3}{10}$ (2) $\pi+\frac{1}{2}$ (3) $(-3)\times\sqrt{7}$

(1) rational (sum of rationals), (2) irrational (rational + irrational), (3) irrational (nonzero rational × irrational). (correct answer)
(1) irrational because fractions make repeating decimals, (2) rational because $\pi$ is close to $3$ , (3) rational because $-3$ cancels the square root.
(1) rational, (2) rational because adding $\frac{1}{2}$ makes $\pi$ repeating, (3) irrational only if $-3$ is irrational.
(1) irrational, (2) irrational, (3) rational because a product is always rational when one factor is an integer.

Explanation: This question tests your understanding of how rational and irrational numbers behave under addition and multiplication—specifically, applying the rules to classify multiple expressions. Let's analyze each expression using our key rules: (1) Rational + rational = rational, always (closure property). (2) Rational + irrational = irrational, always (proven by contradiction). (3) Nonzero rational × irrational = irrational, always (proven by contradiction). For expression (1): 2/5 + 3/10. Both 2/5 and 3/10 are rational (they're fractions with integer numerators and denominators). By closure of rationals under addition, their sum is rational. We can verify: 2/5 + 3/10 = 4/10 + 3/10 = 7/10, which is indeed a ratio of integers, hence rational. For expression (2): π + 1/2. Here π is irrational and 1/2 is rational. By our rule that rational + irrational = irrational, the sum π + 1/2 must be irrational. If it were rational, we could rearrange to get π = (π + 1/2) - 1/2 = rational - rational = rational, contradicting that π is irrational. For expression (3): (-3) × √7. Here -3 is a nonzero rational and √7 is irrational. By our rule that nonzero rational × irrational = irrational, the product (-3)√7 must be irrational. If it were rational, we could divide by -3 to get √7 = ((-3)√7)/(-3) = rational/rational = rational, contradicting that √7 is irrational. Choice A correctly classifies all three: (1) rational (sum of rationals), (2) irrational (rational + irrational), (3) irrational (nonzero rational × irrational), and provides the correct reasoning for each. Choice B makes multiple errors: claims fractions make repeating decimals (true, but repeating decimals are rational!), says π becomes close to 3 when adding 1/2 (nonsense), and thinks -3 'cancels' the square root (multiplication doesn't cancel irrationality). The three key facts to remember: (1) Rational + rational = rational, always. (2) Rational + irrational = irrational, always. (3) Nonzero rational × irrational = irrational, always. These are universal rules you can rely on! Apply them systematically to classify any expression involving rationals and irrationals.

Question 20

The polynomial $s(x) = 2x^3 - 8x^2 + 6x$ can be factored by first removing the common factor. After complete factorization, what information about the graph can be determined from the zeros?

The graph has x-intercepts at $x = 0, 1, 3$ and crosses the x-axis at all three points, with the y-intercept at $(0, 0)$ (correct answer)
The graph has x-intercepts at $x = 0, 1, 3$ but touches without crossing at $x = 0$ due to the factored form having $x^2$
The graph has x-intercepts at $x = 1, 3$ only, since the factor of $x$ doesn't create a visible intercept on the graph
The graph has x-intercepts at $x = 0, 1, 3$ and passes through the origin since one factor is $x$

Explanation: Factoring: $s(x) = 2x^3 - 8x^2 + 6x = 2x(x^2 - 4x + 3) = 2x(x-1)(x-3)$ . The zeros are at $x = 0, 1, 3$ , each appearing once in the factored form, so the graph crosses the x-axis at all three points. The y-intercept is at $(0,0)$ since $s(0) = 0$ . Choice B incorrectly suggests touching behavior at $x = 0$ . Choice C incorrectly excludes $x = 0$ as an intercept. Choice D is partially correct but doesn't fully describe the crossing behavior.

Question 21

Function $a$ is defined by the equation $a(x) = x^2 - 6x + 5$ . The table shows values for function $b$ . Which function has the greater minimum value, and by how much?

Function $a$ has a minimum 3 units greater than the minimum of $b$
Function $b$ has a minimum 3 units greater than the minimum of $a$
Function $a$ has a minimum 1 unit greater than the minimum of $b$
Function $b$ has a minimum 1 unit greater than the minimum of $a$ (correct answer)

Explanation: For a(x) = x¬≤ - 6x + 5, completing the square: a(x) = (x-3)¬≤ - 4, so the minimum is -4. From the table, the minimum value of b appears to be -3 (at x = 2). Since -3 - (-4) = 1, function b has a minimum that is 1 unit greater than the minimum of a.

Question 22

Function $h$ is described verbally as: “a linear function with slope $-4$ and y-intercept $2$ .”

Function $k$ is given by the table:

$x$ : $-1,\ 0,\ 1$

$k(x)$ : $5,\ 3,\ 1$

Which function has the steeper slope (greater slope magnitude)?

Function $h$ , because $|-4|>|-2|$ . (correct answer)
Function $k$ , because its slope is $-1$ .
Function $k$ , because its slope is $-2$ and $|-2|>|-4|$ .
They are equally steep, because both slopes are negative.

Explanation: This question tests your ability to work with functions represented in different ways—like formulas, graphs, and tables—and compare their properties. Functions can be shown in multiple ways: an equation gives you a formula to calculate with, a graph shows the visual pattern, a table lists specific input-output pairs, and a verbal description explains the function in words. Each representation makes different features easy to see—graphs show maximums clearly, formulas make y-intercepts obvious (just set x = 0), and tables give you exact values to read directly. To compare slope magnitudes, note the verbal description gives h a slope of -4 (magnitude 4); for k's table, calculate slope as (3-5)/(0-(-1)) = -2 or (1-3)/(1-0) = -2 (magnitude 2), so 4 > 2. Choice A correctly identifies that function h has the steeper slope because |-4| > |-2|. If you thought k's slope was -1, that's okay—just practice calculating change in y over change in x from table points to get more comfortable. When comparing functions in different forms: (1) Identify what property you're comparing, (2) Extract that property from each representation using the appropriate method (substitute for formulas, read coordinates from graphs, find values in tables), (3) Compare the extracted values. Quick y-intercept trick: in a formula, set x=0 and calculate; on a graph, see where it crosses the y-axis; in a table, find the y-value when x=0—three different methods, same property! Similarly, for comparing slopes of linear functions: read m from y=mx+b, calculate rise/run from a graph, or find Δy/Δx from consecutive table entries.

Question 23

What is $\sqrt[5]{x^3}$ in exponential form?

$x^{5/3}$
$x^{3/5}$ (correct answer)
$x^{15}$
$x^{2/5}$

Explanation: This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Converting between radicals and rational exponents follows a simple pattern: the denominator of the exponent tells you which root (square root is 1/2, cube root is 1/3), and the numerator tells you what power. So ∛(x²) = x^(2/3) because we're taking the cube root (denominator 3) of x squared (numerator 2). For ⁵√(x³), we have the fifth root (denominator 5) of x cubed (numerator 3). This converts directly to x^(3/5). We can verify this: the fifth root is represented by the exponent 1/5, and x³ raised to the 1/5 power gives us x^(3·1/5) = x^(3/5). Choice B is correct because ⁵√(x³) means the fifth root (denominator 5) of x cubed (numerator 3), giving us x^(3/5). Choice A incorrectly inverts the fraction to 5/3, which would represent the cube root of x⁵, not the fifth root of x³. The key to converting: denominator of exponent = which root, numerator = which power. So x^(3/4) means fourth root of x cubed: ⁴√(x³). To remember which is which, think '3 on top means power of 3, 4 on bottom means 4th root.' The fraction tells you everything!

Question 24

A student is solving the equation $3(x - 4) = 2x + 1$ . After distributing on the left side, they write $3x - 12 = 2x + 1$ . What property or principle justifies this step?

The distributive property allows multiplication over subtraction, maintaining equality (correct answer)
The associative property allows regrouping terms while preserving the equation
The commutative property allows rearranging terms without changing the solution
The substitution property allows replacing expressions with equivalent forms

Explanation: The distributive property $a(b - c) = ab - ac$ justifies multiplying 3 by both terms inside the parentheses: $3(x - 4) = 3 \cdot x - 3 \cdot 4 = 3x - 12$ . This maintains equality because we're applying the same operation to the same expression. Choice B (associative) deals with grouping, choice C (commutative) deals with order, and choice D (substitution) is too vague for this specific algebraic manipulation.

Question 25

What is the minimum value of $f(x)=x^2+10x+18$ ? (Use completing the square.)

$7$
$-25$
$-7$ (correct answer)
$25$

Explanation: This question tests your understanding of completing the square to rewrite a quadratic in vertex form, which reveals the maximum or minimum value. The vertex of a parabola is its extreme point: if the parabola opens upward (a > 0), the vertex is the minimum (lowest point), and if it opens downward (a < 0), the vertex is the maximum (highest point). The y-coordinate of the vertex, k, is that extreme value. Starting with f(x) = x² + 10x + 18, we complete the square on the first two terms: take half of 10 to get 5, square it to get 25, then add and subtract inside: f(x) = (x² + 10x + 25) - 25 + 18. The first part is a perfect square (x + 5)², and simplifying the constants: -25 + 18 = -7. So vertex form is f(x) = (x + 5)² - 7, with vertex at (-5, -7)! Choice B is correct because from the vertex form (x + 5)² - 7, we see the minimum value is -7, since the coefficient of the squared term is positive (1), making the parabola open upward. Choice A might tempt you if you confused the perfect square value (25) with the minimum, but remember the minimum value is the y-coordinate of the vertex, which is -7, not the value we used to complete the square. Remember why we're doing this: vertex form isn't just busywork—it instantly tells you the most important point on the parabola (the vertex) without graphing or calculating. In real-world problems, that vertex often represents the best or worst outcome, like maximum profit or minimum cost. Powerful stuff!

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Algebra

Algebra Practice Test: Practice Test 2

Practice Test 2 for Algebra: real questions and explanations from the Varsity Tutors practice-test pool.

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Question 1

$A = (1 + r)^t[P + \frac{500}{(1 + r)^2}]$ , showing a common factor of $(1 + r)^t$ with a bracketed expression
$A = P(1 + r)^t + \frac{500}{(1 + r)^2}(1 + r)^t$ , showing both terms as multiples of $(1 + r)^t$ with different coefficients
$A = (P + 500)(1 + r)^t - 500(1 + r)^2$ , showing the sum distributed across exponential terms
$A = (1 + r)^{t-2}[P(1 + r)^2 + 500]$ , showing a common exponential factor and a bracketed sum depending on $P$ (correct answer)

Question 2

A store sign lists a product weight as 1.50 lb. The scale used in the store reads to the nearest 0.1 lb. Identify the false precision in the statement.

The unit lb is too precise for a scale
The 1.50 suggests precision to the nearest 0.01 lb, which the scale cannot measure (correct answer)
The 1.50 is too rounded; it should be 1.5 exactly
The weight should be reported in ounces only

Question 3

Given $f(x) = x^2 - 4x - 12$ , a student factors this as $(x - 6)(x + 2)$ and identifies key features. Which analysis of the parabola's symmetry is correct?

The parabola is symmetric about the line $x = 4$ passing through the y-intercept
The parabola is symmetric about the line $x = -2$ passing through one zero
The parabola is symmetric about the line $x = 6$ passing through one zero
The parabola is symmetric about the line $x = 2$ passing through the vertex (correct answer)

Question 4

Solve the system: $\begin{cases} 3x-y=7 \\ x+y=5 \end{cases}$ What is the solution $(x, y)$ ?

$(3,2)$ (correct answer)
$(2,3)$
$(4,1)$
$(1,4)$

Question 5

Where is the vertex of $g(x)=|x-3|+2$ ?

$(2,3)$
$(3,-2)$
$(3,2)$ (correct answer)
$(-3,2)$

Question 6

A community garden is planting tomatoes and peppers. Let $x$ = number of tomato plants and $y$ = number of pepper plants.

Constraints:

There is space for at most 30 plants total: $x+y\le 30$ .
Tomato plants must be at least 10: $x\ge 10$ .
Each tomato plant needs 2 minutes/day of watering and each pepper plant needs 3 minutes/day; they have at most 75 minutes/day: $2x+3y\le 75$ .
$x$ and $y$ must be nonnegative whole numbers.

Is the point $(12, 6)$ a viable solution? Explain by identifying whether it satisfies all constraints.

Viable; it satisfies $x+y\le 30$ , $x\ge 10$ , $2x+3y\le 75$ , and uses nonnegative whole numbers. (correct answer)
Nonviable; it violates the space constraint because $12+6>30$ .
Nonviable; it violates the watering constraint because $2(12)+3(6)>75$ .
Nonviable; it violates the minimum tomato constraint because $12<10$ .

Question 7

A parabola is defined by $h(x)=-(x-4)^2+1$ . What is the maximum value of $h(x)$ ?

$-1$
$4$
$5$
$1$ (correct answer)

Question 8

The function $p(x)=\sqrt{x}$ assigns one output to each input in its domain. What is the range of $p$ ? (Answer in interval notation.)

$(-\infty,\infty)$
$[0,\infty)$ (correct answer)
$(-\infty,0]$
$(0,\infty)$

Question 9

If $\sqrt[3]{x^2} = x^{\frac{a}{b}}$ where $\frac{a}{b}$ is in lowest terms, what is the value of $a + b$ ?

$8$
$3$
$6$
$5$ (correct answer)

Question 10

For the linear function $p(x)=4x-7$ , what is the average rate of change of $p(x)$ from $x=-2$ to $x=3$ ?

$-4$
$4$ (correct answer)
$\frac{4}{5}$
$20$

Question 11

Divide and simplify (write your answer as a simplified rational expression): $\frac{x^2-4}{x+1}\div\frac{x+2}{x+1}$

$\frac{x^2-4}{x+2}$
$\frac{x-2}{x+1}$
$\frac{x+2}{x^2-4}$
$x-2$ (correct answer)

Question 12

A quadratic equation of the form $x^2 + bx + c = 0$ has solutions $x = 3 \pm 2i$ . What is the value of $b + c$ ?

$19$
$13$
$7$ (correct answer)
$-7$

Question 13

The profit $P$ (in dollars) from selling $x$ items is given by $P(x) = -3x^2 + 24x - 36$ . How many items should be sold to maximize profit, and what is the maximum profit?

4 items; $12 profit (correct answer)
8 items; $60 profit
4 items; $-36 profit
12 items; $108 profit

Question 14

If $g(x)=x^2+3x-4$ , what is $g(-2)$ ?

$-6$ (correct answer)
$-14$
$6$
$-2$

Question 15

Let $f(x)=2x$ and $g(x)=x^2+1$ . What is $f(g(3))$ ?

$20$ (correct answer)
$19$
$8$
$2(3^2)+1=19$

Question 16

Two students complete the square for the same equation $x^2 - 8x + 7 = 0$ . Student A writes $(x - 4)^2 = 9$ and Student B writes $(x - 4)^2 - 9 = 0$ . Which statement about their work is correct?

Only Student B's form is correct because all terms should be on one side
Only Student A's form is correct because it matches the required format
Both forms are mathematically equivalent and correct for solving the equation (correct answer)
Both students made errors because the correct form should be $(x + 4)^2 = 9$

Question 17

Evaluate $27^{2/3}$ .

$6$
$12$
$9$ (correct answer)
$18$

Question 18

Rewrite $16^{3/4}$ using radicals, then evaluate.

$12$
$8$ (correct answer)
$4$
$6$

Explanation: This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Rational exponents give us an exponential way to write roots: $x^{1/n}$ means the nth root of x (like $x^{1/2} = \sqrt{x}$ and $x^{1/3} = \sqrt[3]{x}$ ), and more generally, $x^{m/n}$ means take the nth root of x, then raise it to the mth power (or do the power first, then the root—either order works!). To evaluate $16^{3/4}$ , first convert to radical form: $16^{3/4} = \sqrt[4]{16^3}$ or $(\sqrt[4]{16})^3$ . Since $\sqrt[4]{16} = 2$ (because $2^4 = 16$ ), we get $2^3 = 8$ . Choice C is correct because $16^{3/4} = (\sqrt[4]{16})^3 = 2^3 = 8$ . Choice B gives 12, which might come from incorrectly calculating 16 × 3/4 = 12, but that's not how rational exponents work. For simplifying with exponent properties, convert all radicals to exponential form first if they aren't already: $\sqrt{x}$ becomes $x^{1/2}$ , $\sqrt[3]{x}$ becomes $x^{1/3}$ , etc. Then use your exponent rules (add when multiplying, subtract when dividing, multiply when doing power of a power). Finally, convert back to radical form if that's what's asked for!

Question 19

Classify each expression as rational or irrational and choose the option that correctly justifies all three:

(1) $\frac{2}{5}+\frac{3}{10}$ (2) $\pi+\frac{1}{2}$ (3) $(-3)\times\sqrt{7}$

(1) rational (sum of rationals), (2) irrational (rational + irrational), (3) irrational (nonzero rational × irrational). (correct answer)
(1) irrational because fractions make repeating decimals, (2) rational because $\pi$ is close to $3$ , (3) rational because $-3$ cancels the square root.
(1) rational, (2) rational because adding $\frac{1}{2}$ makes $\pi$ repeating, (3) irrational only if $-3$ is irrational.
(1) irrational, (2) irrational, (3) rational because a product is always rational when one factor is an integer.

Question 20

The polynomial $s(x) = 2x^3 - 8x^2 + 6x$ can be factored by first removing the common factor. After complete factorization, what information about the graph can be determined from the zeros?

The graph has x-intercepts at $x = 0, 1, 3$ and crosses the x-axis at all three points, with the y-intercept at $(0, 0)$ (correct answer)
The graph has x-intercepts at $x = 0, 1, 3$ but touches without crossing at $x = 0$ due to the factored form having $x^2$
The graph has x-intercepts at $x = 1, 3$ only, since the factor of $x$ doesn't create a visible intercept on the graph
The graph has x-intercepts at $x = 0, 1, 3$ and passes through the origin since one factor is $x$

Question 21

Function $a$ is defined by the equation $a(x) = x^2 - 6x + 5$ . The table shows values for function $b$ . Which function has the greater minimum value, and by how much?

Function $a$ has a minimum 3 units greater than the minimum of $b$
Function $b$ has a minimum 3 units greater than the minimum of $a$
Function $a$ has a minimum 1 unit greater than the minimum of $b$
Function $b$ has a minimum 1 unit greater than the minimum of $a$ (correct answer)

Question 22

Function $h$ is described verbally as: “a linear function with slope $-4$ and y-intercept $2$ .”

Function $k$ is given by the table:

$x$ : $-1,\ 0,\ 1$

$k(x)$ : $5,\ 3,\ 1$

Which function has the steeper slope (greater slope magnitude)?

Function $h$ , because $|-4|>|-2|$ . (correct answer)
Function $k$ , because its slope is $-1$ .
Function $k$ , because its slope is $-2$ and $|-2|>|-4|$ .
They are equally steep, because both slopes are negative.

Question 23

What is $\sqrt[5]{x^3}$ in exponential form?

$x^{5/3}$
$x^{3/5}$ (correct answer)
$x^{15}$
$x^{2/5}$

Explanation: This question tests your understanding of the connection between radicals and rational exponents, and how to rewrite expressions using exponent properties. Converting between radicals and rational exponents follows a simple pattern: the denominator of the exponent tells you which root (square root is 1/2, cube root is 1/3), and the numerator tells you what power. So ∛(x²) = x^(2/3) because we're taking the cube root (denominator 3) of x squared (numerator 2). For ⁵√(x³), we have the fifth root (denominator 5) of x cubed (numerator 3). This converts directly to x^(3/5). We can verify this: the fifth root is represented by the exponent 1/5, and x³ raised to the 1/5 power gives us x^(3·1/5) = x^(3/5). Choice B is correct because ⁵√(x³) means the fifth root (denominator 5) of x cubed (numerator 3), giving us x^(3/5). Choice A incorrectly inverts the fraction to 5/3, which would represent the cube root of x⁵, not the fifth root of x³. The key to converting: denominator of exponent = which root, numerator = which power. So x^(3/4) means fourth root of x cubed: ⁴√(x³). To remember which is which, think '3 on top means power of 3, 4 on bottom means 4th root.' The fraction tells you everything!

Question 24

A student is solving the equation $3(x - 4) = 2x + 1$ . After distributing on the left side, they write $3x - 12 = 2x + 1$ . What property or principle justifies this step?

The distributive property allows multiplication over subtraction, maintaining equality (correct answer)
The associative property allows regrouping terms while preserving the equation
The commutative property allows rearranging terms without changing the solution
The substitution property allows replacing expressions with equivalent forms

Question 25

What is the minimum value of $f(x)=x^2+10x+18$ ? (Use completing the square.)

$7$
$-25$
$-7$ (correct answer)
$25$