Math 1 Quiz: Selecting Efficient Strategies

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Selecting Efficient StrategiesQuestion 1 of 20

To solve $|2x - 3| = |x + 1|$ , which approach would minimize the number of cases to consider?

Square both sides to eliminate absolute value signs, then solve the resulting quadratic equation and check all solutions

Consider all four possible sign combinations: both expressions positive, both negative, first positive and second negative, first negative and second positive

Set up two cases:

2x - 3 = x + 1

and

2x - 3 = -(x + 1)

, then solve each equation and verify solutions

Find the critical points where each expression inside absolute values equals zero, then test intervals between these points

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Math 1 Quiz

Math 1 Quiz: Selecting Efficient Strategies

Practice Selecting Efficient Strategies in Math 1 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Selecting Efficient Strategies, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 1.

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Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

To solve $|2x - 3| = |x + 1|$ , which approach would minimize the number of cases to consider?

Square both sides to eliminate absolute value signs, then solve the resulting quadratic equation and check all solutions
Consider all four possible sign combinations: both expressions positive, both negative, first positive and second negative, first negative and second positive
Set up two cases: $2x - 3 = x + 1$ and $2x - 3 = -(x + 1)$ , then solve each equation and verify solutions (correct answer)
Find the critical points where each expression inside absolute values equals zero, then test intervals between these points

Explanation: The equation

|A| = |B|

is equivalent to

A = B

A = -B

. So

2x - 3 = x + 1

gives

x = 4

, and

2x - 3 = -(x + 1)

gives

3x = 2

, so

x = \frac{2}{3}

. Both solutions check out. This method directly uses the property of absolute value equations. Choice A works but requires checking for extraneous solutions. Choice B considers unnecessary cases since

|A| = |B|

only requires two cases, not four. Choice D is more complex than needed for this equation type.

Question 2

A student needs to find the value of $x$ in the equation $3(x + 4) = 2x + 15$ . She considers three approaches: (1) Distribute first, then solve; (2) Divide both sides by 3 first, then solve; (3) Subtract $2x$ from both sides first, then solve. Which strategy is most efficient and why?

Strategy (1) because distributing eliminates parentheses immediately, leading to a standard linear equation in fewest steps (correct answer)
Strategy (2) because dividing by 3 first simplifies the coefficients and avoids working with larger numbers throughout
Strategy (3) because isolating variable terms on one side first reduces the equation to its simplest form immediately
All three strategies are equally efficient since they require the same total number of algebraic operations to reach the solution

Explanation: Strategy (1) is most efficient. Distributing gives

3x + 12 = 2x + 15

, then subtracting

2x

gives

x + 12 = 15

, then subtracting 12 gives

x = 3

. This takes 3 clear steps. Strategy (2) creates fractions unnecessarily:

x + 4 = \frac{2x + 15}{3}

, requiring fraction arithmetic. Strategy (3) leads to

3(x + 4) - 2x = 15

, which still requires distributing the 3, making it longer than strategy (1).

Question 3

A student wants to determine if $x = 2$ is a solution to the system: $\begin{cases} 3x + y = 8 \\ x - 2y = -6 \end{cases}$ . She considers: (1) Substitute $x = 2$ into both equations and solve for $y$ ; (2) Solve the system completely first, then check if the solution matches; (3) Graph both equations and see if they intersect at $x = 2$ . Which strategy most efficiently answers the specific question?

Strategy (1) because it directly tests the given value without unnecessary computation of the complete solution (correct answer)
Strategy (2) because finding the complete solution provides more information and confirms the answer definitively
Strategy (3) because graphing provides visual confirmation and helps identify any errors in algebraic manipulation
Strategy (2) and (3) together because algebraic and graphical methods should always be used to verify solutions

Explanation: Strategy (1) is most efficient for this specific question. Substituting

x = 2

: First equation gives

6 + y = 8

, so

y = 2

. Second equation gives

2 - 2y = -6

, so

y = 4

. Since we get different

y

-values,

x = 2

is not a solution. This answers the question directly. Strategy (2) requires solving the entire system unnecessarily. Strategy (3) requires graphing, which is time-consuming when only testing one specific value.

Question 4

To find the intersection points of $y = x^2 - 4x + 3$ and $y = -x + 3$ , a student considers: (1) Set the equations equal and solve the resulting quadratic; (2) Complete the square on the parabola first, then find intersections; (3) Use substitution to eliminate $y$ , then solve for $x$ . Which approach demonstrates the most efficient reasoning?

Strategy (3) because substitution method provides a more organized approach to solving systems of equations consistently
Strategy (2) because completing the square reveals the vertex form, making intersection analysis more systematic and visual
Strategy (1) because setting equations equal directly creates the condition for intersection without unnecessary algebraic manipulation (correct answer)
Strategy (2) and (3) are equivalent in efficiency since both involve the same underlying algebraic transformations

Explanation: When finding intersection points of two curves, you're looking for where their

y

-values are equal at the same

x

-coordinates. This means the most direct approach is to set the equations equal to each other. Setting

x^2 - 4x + 3 = -x + 3

immediately gives you the condition for intersection. Rearranging yields

x^2 - 3x = 0

, which factors as

x(x - 3) = 0

, so

x = 0

x = 3

. You can then substitute back to find the

y

-coordinates. This approach cuts straight to what you need without extra steps. Choice A suggests substitution is more organized, but substitution (replacing

y

with one expression in the other equation) actually leads to the exact same equation as setting them equal—it's just a longer way to get there. Choice B claims completing the square first makes the analysis more systematic. While completing the square gives you

y = (x - 2)^2 - 1

, this vertex form doesn't actually simplify finding intersections. You still need to set the equations equal, so you've added an unnecessary step. Choice D incorrectly states that strategies 2 and 3 are equivalent in efficiency. Strategy 2 includes the extra step of completing the square, while strategy 3 (substitution) is equivalent to strategy 1 but written differently. Study tip: For intersection problems, always ask yourself: "What's the most direct path to where the functions are equal?" Usually, it's setting them equal immediately rather than transforming first.

Question 5

To determine the domain of $f(x) = \frac{\sqrt{x - 2}}{x^2 - 9}$ , a student could: (1) Find where the numerator is defined, then exclude zeros of denominator; (2) Graph the function and observe where it's undefined; (3) Test specific values systematically to identify problematic regions. Which analytical approach is most comprehensive and efficient?

Strategy (1) combined with (2) because algebraic analysis should always be verified graphically for complete understanding
Strategy (2) because graphing reveals discontinuities and undefined regions more clearly than algebraic analysis alone
Strategy (3) because testing specific values provides concrete examples of where the function fails to exist
Strategy (1) because it systematically identifies all restrictions by examining numerator and denominator requirements separately (correct answer)

Explanation: When finding the domain of a rational function with a square root, you need to systematically identify all restrictions by examining both the numerator and denominator separately. This algebraic approach ensures you don't miss any constraints. For

f(x) = \frac{\sqrt{x - 2}}{x^2 - 9}

, strategy (1) works methodically: First, the square root requires

x - 2 \geq 0

, so

x \geq 2

. Then, the denominator cannot equal zero, so

x^2 - 9 \neq 0

, meaning

x \neq \pm 3

. Since we already need

x \geq 2

, we only exclude

x = 3

from our domain. The result is

x \geq 2

and

x \neq 3

, or

[2, 3) \cup (3, \infty)

. Option A suggests combining algebraic and graphical methods, but while graphing can verify results, it's not necessary for completeness when the algebraic analysis is thorough. Option B overvalues graphing—visual methods can miss subtle restrictions or provide imprecise boundaries, especially for square root domains. Option C relies on testing specific values, which is inefficient and likely to miss critical points since you'd need to test infinitely many values to be certain. Option D correctly identifies that the systematic algebraic approach finds all restrictions efficiently. By separately analyzing numerator and denominator requirements, you guarantee complete coverage of all domain restrictions without redundancy. For rational functions involving radicals, always start with the algebraic approach: identify radical restrictions first, then find denominator zeros, then combine the constraints logically.

Question 6

To find the maximum value of $f(x) = -2x^2 + 8x - 3$ on the interval $[0, 5]$ , a student considers: (1) Find the vertex using $x = -\frac{b}{2a}$ , then check if it's in the interval; (2) Complete the square to rewrite in vertex form; (3) Evaluate the function at the endpoints and critical points. Which approach most directly addresses the constraint of the given interval?

Strategy (1) because finding the vertex directly locates the maximum of the parabola, then verifies interval constraints
Strategy (2) because vertex form immediately reveals the maximum value and its location for comparison with interval boundaries
Strategy (3) because it systematically evaluates all candidate points where the maximum could occur within the constraint (correct answer)
Strategy (1) and (2) are equivalent in efficiency since both locate the vertex, but (3) requires more computation

Explanation: Strategy (3) is most direct for constrained optimization. The vertex is at

x = -\frac{8}{2(-2)} = 2

, which is in

[0,5]

. We must check:

f(0) = -3

f(2) = -8 + 16 - 3 = 5

f(5) = -50 + 40 - 3 = -13

. Maximum is 5 at

x = 2

. Strategy (3) explicitly addresses the interval constraint by checking all candidates. Strategies (1) and (2) find the vertex but still require endpoint checking since we're on a restricted interval, making (3) more systematic.

Question 7

A student needs to simplify $\frac{x^2 - 5x + 6}{x^2 - 4} \cdot \frac{x + 2}{x - 3}$ . She considers: (1) Factor all polynomials first, then multiply and cancel; (2) Multiply the fractions first, then factor the resulting expression; (3) Convert to a single fraction, then factor numerator and denominator. Which sequence of operations minimizes algebraic manipulation?

Strategy (2) and (3) are equally efficient since both avoid premature factoring of individual terms
Strategy (2) because multiplying first consolidates the expression into a single step, then factoring handles the simplified result
Strategy (3) because working with a single fraction provides clearer structure for identifying and canceling common factors
Strategy (1) because factoring before multiplication reveals common factors immediately, allowing cancellation without complex expansion (correct answer)

Explanation: When simplifying complex rational expressions involving multiplication, the order of operations significantly impacts the amount of algebraic work required. The key principle is to identify and eliminate common factors as early as possible to avoid unnecessary multiplication and expansion. Let's examine strategy (1): factoring first. The expression becomes

\frac{(x-2)(x-3)}{(x-2)(x+2)} \cdot \frac{x+2}{x-3}

. Now you can immediately see that

(x-2)

(x-3)

, and

(x+2)

appear in both numerators and denominators, allowing instant cancellation to get

1

. This requires minimal computation. Compare this to strategy (2): multiplying first gives

\frac{(x^2-5x+6)(x+2)}{(x^2-4)(x-3)}

. You'd need to expand the numerator to

x^3-3x^2-4x+12

and the denominator to

x^3-3x^2-4x+12

, only to discover they're identical. This involves much more algebraic manipulation. Strategy (3) creates the same computational burden as strategy (2), since you still need to multiply before factoring. Answer A is wrong because strategies (2) and (3) actually require extensive multiplication and expansion. Answer B is incorrect because multiplying first creates a more complex expression, not a simpler one. Answer C is flawed because the "single fraction" approach doesn't eliminate the need for complex multiplication—it actually necessitates it. Study tip: Always factor rational expressions completely before multiplying. Common factors become visible immediately, allowing cancellation that eliminates most of the computational work. Think "factor first, multiply what's left."

Question 8

To solve the system $\begin{cases} 2x + 3y = 7 \\ 4x + 6y = 14 \end{cases}$ , a student notices the second equation and considers: (1) Use elimination method by multiplying the first equation by -2; (2) Recognize that the second equation is a multiple of the first; (3) Use substitution by solving the first equation for $x$ . Which approach demonstrates the most efficient mathematical reasoning?

Strategy (1) because elimination is the standard systematic approach for solving linear systems with two variables
Strategy (2) because recognizing the relationship between equations immediately identifies the system as dependent with infinitely many solutions (correct answer)
Strategy (3) because substitution method provides a more direct path to finding specific solution values
Strategy (1) and (3) are equally valid since both lead to the same solution set through different systematic approaches

Explanation: Strategy (2) is most efficient. Observing that

4x + 6y = 14

is exactly

2(2x + 3y) = 2(7)

, we immediately recognize this is the same equation written twice. The system has infinitely many solutions along the line

2x + 3y = 7

. This insight saves all computational work. Strategy (1) would lead to

0 = 0

, confirming dependence but requiring more steps. Strategy (3) misses the key insight and attempts to find a unique solution that doesn't exist. Recognizing structural relationships is more efficient than blind computation.

Question 9

To evaluate $\sqrt{48} + \sqrt{75} - \sqrt{27}$ , a student could: (1) Use a calculator to find decimal approximations; (2) Simplify each radical by factoring out perfect squares first; (3) Convert each to exponential form using fractional exponents. Which approach is most mathematically efficient for obtaining an exact answer?

Strategy (1) because calculators eliminate computational errors and provide immediate numerical results for comparison
Strategy (2) because factoring perfect squares allows combination of like radicals, yielding the simplest exact form (correct answer)
Strategy (3) because exponential notation provides a more systematic approach to radical arithmetic operations
Strategy (1) and (2) are equally efficient since both avoid the complexity of exponential form manipulation

Explanation: Strategy (2) is most efficient for exact answers. Simplifying:

\sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3}

\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}

\sqrt{27} = \sqrt{9 \cdot 3} = 3\sqrt{3}

. So the expression becomes

4\sqrt{3} + 5\sqrt{3} - 3\sqrt{3} = 6\sqrt{3}

. Strategy (1) gives approximations, not exact answers. Strategy (3) makes the problem unnecessarily complex with fractional exponents.

Question 10

A student needs to solve $\frac{2x - 3}{4} - \frac{x + 1}{6} = 1$ . She considers: (1) Find common denominator 12, then solve; (2) Multiply entire equation by 12 to clear fractions first; (3) Convert each fraction to decimal form, then solve. Which method minimizes computational complexity while maintaining accuracy?

Strategy (1) because working with a common denominator maintains fraction form throughout and follows standard procedures
Strategy (2) because clearing fractions immediately eliminates denominators and converts to a simpler linear equation (correct answer)
Strategy (3) because decimal conversion avoids fraction arithmetic entirely and makes calculations more straightforward
Strategy (1) and (2) are equally efficient since both methods preserve exactness while simplifying the equation structure

Explanation: Strategy (2) is most efficient. Multiplying by 12:

12 \cdot \frac{2x - 3}{4} - 12 \cdot \frac{x + 1}{6} = 12 \cdot 1

gives

3(2x - 3) - 2(x + 1) = 12

, then

6x - 9 - 2x - 2 = 12

, so

4x = 23

and

x = \frac{23}{4}

. This avoids fraction arithmetic throughout. Strategy (1) requires working with twelfths unnecessarily. Strategy (3) introduces rounding errors and decimals like 0.333... that complicate calculations.

Question 11

To solve $\frac{3}{x-1} + \frac{2}{x+1} = \frac{4}{x^2-1}$ , which method would avoid the most computational complexity?

Multiply through by the least common denominator $(x^2-1)$ immediately, then solve the resulting linear equation while noting restrictions (correct answer)
First rewrite each fraction with the common denominator $x^2-1$ , then set the numerators equal and solve the linear equation
Cross multiply after combining the fractions on the left side into a single fraction, then solve the resulting proportion
Use partial fraction decomposition on the right side, then match coefficients with the left side to find the value of $x$

Explanation: Since

x^2-1 = (x-1)(x+1)

, multiplying by

x^2-1

gives

3(x+1) + 2(x-1) = 4

, which simplifies to

3x + 3 + 2x - 2 = 4

, so

5x = 3

and

x = \frac{3}{5}

. We must check that

x \neq \pm 1

(satisfied). This method eliminates fractions immediately. Choice B involves the same steps but with additional fraction manipulation. Choice C requires combining fractions first, adding steps. Choice D reverses the natural direction since partial fractions are used to decompose, not compose.

Question 12

To find the maximum value of $f(x) = -2x^2 + 8x - 3$ on the interval $[0, 5]$ , which strategy would be most direct?

Take the derivative, set it equal to zero, check if critical points lie in the interval, then compare with endpoint values
Complete the square to rewrite in vertex form, then determine if the vertex lies within the given interval (correct answer)
Evaluate the function at several points within the interval and choose the point that gives the highest value
Use the quadratic formula to find the zeros, then evaluate the function at the midpoint between zeros

Explanation: Completing the square gives

f(x) = -2(x-2)^2 + 5

, immediately revealing that the maximum is 5 at

x = 2

. Since

2 \in [0,5]

, this is the global maximum on the interval. This method directly provides both the maximum value and its location. Choice A works but requires calculus and additional steps. Choice C is inefficient and doesn't guarantee finding the true maximum. Choice D finds zeros but doesn't directly lead to the maximum value.

Question 13

A function $f(x) = ax^3 + bx^2 + cx + d$ passes through points $(0, 5)$ , $(1, 8)$ , $(2, 15)$ , and $(-1, 0)$ . Which method would most efficiently determine the coefficients?

Set up the system of four equations in four unknowns and solve using Gaussian elimination with full matrix operations
Use the point $(0, 5)$ to find $d$ immediately, then solve the remaining three-equation system for $a$ , $b$ , and $c$ (correct answer)
Apply Lagrange interpolation formula directly to construct the polynomial without solving for individual coefficients
Use finite differences to identify the pattern in y-values and construct the polynomial from the difference table

Explanation: Since

f(0) = d = 5

, we immediately know

d

without calculation. This reduces the problem to a

3 \times 3

system instead of

4 \times 4

, significantly reducing computational work. The remaining system becomes:

a + b + c = 3

8a + 4b + 2c = 10

-a + b - c = -5

. Choice A unnecessarily solves a larger system. Choice C works but is computationally intensive for this problem size. Choice D requires constructing a difference table and may not clearly reveal the cubic pattern with only four points.

Question 14

To solve the system $$ \begin{cases} 2x + 3y = 7 \ 6x + 9y = 21 \end{cases}

Use substitution method by solving the first equation for $x$ in terms of $y$ , then substituting into the second equation
Apply elimination method by multiplying the first equation by -3 and adding it to the second equation
Recognize that the second equation is 3 times the first equation, indicating infinitely many solutions along the line $2x + 3y = 7$ (correct answer)
Use Cramer's rule to find the values of $x$ and $y$ by calculating the appropriate determinant ratios

Explanation: Observing that

6x + 9y = 21

is exactly

3(2x + 3y) = 3(7)

immediately reveals that these are the same line, so the system has infinitely many solutions. This recognition eliminates all computational work. Choice A leads to

0 = 0

, requiring additional work to interpret. Choice B also leads to

0 = 0

after elimination steps. Choice D fails because the determinant is zero, requiring recognition that the system is dependent anyway.

Question 15

To evaluate $\frac{2^{10} \cdot 5^{10}}{4^4 \cdot 25^3}$ , which approach would minimize calculation time while maintaining accuracy?

Calculate each exponential term separately using repeated multiplication, then perform the division
Convert everything to the same base (base 2) and use exponent rules to simplify before calculating
Rewrite as $\frac{(2 \cdot 5)^{10}}{(2^2)^4 \cdot (5^2)^3}$ and simplify using exponent rules before evaluating (correct answer)
Use logarithms to convert the expression to addition and subtraction, then convert back using antilogarithms

Explanation: Rewriting gives

\frac{10^{10}}{2^8 \cdot 5^6} = \frac{2^{10} \cdot 5^{10}}{2^8 \cdot 5^6} = 2^2 \cdot 5^4 = 4 \cdot 625 = 2500

. This uses the structure of the expression efficiently. Choice A involves unnecessary large number calculations. Choice B requires converting 5 and 25 to base 2, creating messy fractional exponents. Choice D adds unnecessary complexity with logarithmic calculations for what can be solved algebraically.

Question 16

To determine if the relation $\{(1,2), (2,3), (3,4), (2,5), (4,1)\}$ is a function, which method would provide the quickest verification?

Plot all ordered pairs on a coordinate plane and apply the vertical line test to check for function properties
List all x-values and y-values separately, then verify that the number of unique x-values equals the total number of ordered pairs
Create a mapping diagram showing arrows from domain elements to range elements and verify one-to-one correspondence
Check if each x-value in the domain corresponds to exactly one y-value by examining the ordered pairs systematically (correct answer)

Explanation: When determining if a relation is a function, you need to verify that each input (x-value) maps to exactly one output (y-value). This is the fundamental definition of a function. Looking at the given relation

\{(1,2), (2,3), (3,4), (2,5), (4,1)\}

, the most efficient approach is to systematically examine each x-value and count how many y-values it pairs with. Scanning through: x = 1 appears once (with y = 2), x = 2 appears twice (with y = 3 and y = 5), x = 3 appears once (with y = 4), and x = 4 appears once (with y = 1). Since x = 2 corresponds to two different y-values, this relation is not a function. This direct examination gives you the answer immediately. Option A, the vertical line test, would work but requires plotting all points first, making it slower than direct inspection. Option B contains a logical error—the number of unique x-values doesn't need to equal the total number of ordered pairs for a function. In fact, a function can have repeated y-values. Option C suggests checking for one-to-one correspondence, but this tests whether a function is one-to-one (injective), not whether the relation is a function at all. For function verification questions, always start by checking if any x-value appears more than once with different y-values. This direct approach saves time and immediately reveals whether the fundamental function requirement is met.

Question 17

To factor $6x^3 - 9x^2 - 6x + 9$ completely, which approach would lead to the solution most quickly?

Use the rational root theorem to find potential rational roots, test each systematically using synthetic division until a root is found
Look for patterns by rearranging as $6x^3 - 6x - 9x^2 + 9$ and factoring by grouping in a different way
Apply the general cubic formula directly to find all three roots, then convert back to factored form using the roots
Group terms strategically as $(6x^3 - 9x^2) + (-6x + 9)$ and factor out common factors from each group (correct answer)

Explanation: When you encounter a four-term polynomial like this, your first instinct should be to try factoring by grouping. This technique works by strategically pairing terms so you can factor out common factors from each pair, then look for a common binomial factor. Option D shows the most efficient approach: grouping as

(6x^3 - 9x^2) + (-6x + 9)

. From the first group, you can factor out

3x^2

3x^2(2x - 3)

. From the second group, you factor out

-3

-3(2x - 3)

. Now you have

3x^2(2x - 3) - 3(2x - 3)

, which factors as

(2x - 3)(3x^2 - 3) = (2x - 3) \cdot 3(x^2 - 1) = 3(2x - 3)(x + 1)(x - 1)

. Option A would eventually work but requires unnecessary computation. Testing rational roots like

\pm 1, \pm 3, \pm \frac{1}{2}

through synthetic division is time-consuming when grouping gives you the answer immediately. Option B suggests a different grouping arrangement, but

6x^3 - 6x - 9x^2 + 9

doesn't create pairs with obvious common factors, making this approach more difficult. Option C is completely impractical. The cubic formula is extremely complex and would take far longer than any other method, even if you could remember it perfectly. Study tip: When you see a four-term polynomial, always try factoring by grouping first. Look for ways to pair terms so each pair has a common factor—this is usually the fastest path to factoring completely.

Question 18

To find the domain of $f(x) = \frac{\sqrt{x-1}}{x^2-5x+6}$ , which sequence of steps would be most systematic?

First identify where the square root is defined, then find where the denominator equals zero, and take the intersection of valid regions
Test several values of $x$ in the function to determine which ones produce real number outputs versus undefined results
Graph the function using technology and observe where the function appears to be undefined or has asymptotes
Factor the denominator completely, identify all restrictions, then determine where the square root expression is defined, combining all constraints (correct answer)

Explanation: When finding the domain of a rational function with a square root, you need to systematically identify all restrictions and combine them properly. The domain consists of all x-values where the function is defined. Option D provides the most systematic approach. First, factor the denominator:

x^2-5x+6 = (x-2)(x-3)

. This immediately shows the denominator equals zero when

x = 2

x = 3

, so these values must be excluded. Next, for the square root

\sqrt{x-1}

to be defined, you need

x-1 \geq 0

, which means

x \geq 1

. Combining these constraints: the domain is

x \geq 1

except

x \neq 2

and

x \neq 3

, or

[1,2) \cup (2,3) \cup (3,\infty)

. Option A has the right components but suggests taking an "intersection of valid regions," which is imprecise language that could lead to errors when combining the restrictions. Option B (testing values) is inefficient and unreliable. You might miss critical boundary points or asymptotes, and testing can't definitively establish the complete domain. Option C (graphing with technology) can be helpful for visualization but isn't systematic for finding exact domain restrictions. Technology might not clearly show removable discontinuities or might have resolution limitations near critical points. For rational functions with radicals, always: factor denominators completely first, identify zeros to exclude, then handle radical restrictions, and finally combine all constraints carefully. This systematic algebraic approach ensures you don't miss any restrictions.

Question 19

To evaluate $\log_2(64) + \log_2(16) - \log_2(32)$ , which approach would be most efficient?

Convert each logarithm to natural logarithms using the change of base formula, then evaluate using a calculator
Use logarithm properties to combine into a single logarithm $\log_2\left(\frac{64 \cdot 16}{32}\right)$ , then evaluate
Recognize that each argument is a power of 2 and convert directly: $\log_2(2^6) + \log_2(2^4) - \log_2(2^5)$ (correct answer)
Apply the definition of logarithm by finding what power of 2 gives each argument, then solve the resulting arithmetic expression

Explanation: Since

64 = 2^6

16 = 2^4

, and

32 = 2^5

, we get

\log_2(2^6) + \log_2(2^4) - \log_2(2^5) = 6 + 4 - 5 = 5

. This uses the property

\log_b(b^x) = x

directly. Choice B works but requires additional calculation:

\frac{64 \cdot 16}{32} = \frac{1024}{32} = 32 = 2^5

, so

\log_2(32) = 5

. Choice A unnecessarily complicates with change of base. Choice D describes the same process as C but less efficiently.

Question 20

A student needs to find the value of $x$ in the equation $3x^2 - 12x + 9 = 0$ . Which strategy would be most efficient and why?

Use the quadratic formula because it works for all quadratic equations and requires no factoring skills
Factor out the common factor of 3 first, then use factoring or completing the square on the simplified equation (correct answer)
Complete the square directly on the original equation to avoid potential arithmetic errors from factoring
Graph the function $y = 3x^2 - 12x + 9$ and find the x-intercepts to get exact decimal values

Explanation: Factoring out the common factor of 3 first gives

3(x^2 - 4x + 3) = 0

, which simplifies to

x^2 - 4x + 3 = 0

. This factors easily as

(x-1)(x-3) = 0

, giving

x = 1

x = 3

. This is most efficient because it reduces computational complexity. Choice A works but involves unnecessary complex arithmetic. Choice C involves more steps with fractions. Choice D is impractical for exact solutions and doesn't utilize the equation's structure.