Rewriting to Reveal Features - Math 2

The profit function for a company is given by $P(x) = -2x^2 + 16x - 24$ where $x$ represents thousands of units sold. To find the number of units that must be sold to break even (profit = 0), the function should be rewritten in which form?
Answer: $P(x) = -2(x - 2)(x - 6)$ to identify the zeros directly To find break-even points where profit equals zero, we need to solve P(x) = 0. The most direct approach is to factor the quadratic to reveal its zeros. The factored form -2(x - 2)(x - 6) immediately shows that P(x) = 0 when x = 2 or x = 6, meaning break-even occurs at 2,000 and 6,000 units. Choice A factors out the leading coefficient but doesn't reveal zeros clearly. Choice C gives vertex form which shows maximum profit but requires additional steps to find zeros. Choice D rearranges terms but doesn't simplify finding zeros.
A quadratic function has the form $g(x) = a(x - 3)^2 + 7$ where $a < 0$ . If this function has zeros, what must be true about the value of $|a|$ ?
Answer: $|a| > \frac{7}{9}$ When you encounter a quadratic function in vertex form with conditions about zeros, you need to connect the vertex location to where the parabola can intersect the x-axis. The function $g(x) = a(x - 3)^2 + 7$ has vertex at $(3, 7)$ and opens downward since $a < 0$ . For this parabola to have zeros (x-intercepts), it must dip low enough from its vertex at height 7 to actually cross the x-axis. To find when zeros exist, set $g(x) = 0$ : $0 = a(x - 3)^2 + 7$ $a(x - 3)^2 = -7$ $(x - 3)^2 = \frac{-7}{a}$ For real solutions to exist, the right side must be positive, which means $\frac{-7}{a} > 0$ . Since $a < 0$ , this inequality holds when $|a| > 7$ . Wait—let me recalculate this more carefully. Since $a < 0$ , we have $a = -|a|$ , so: $(x - 3)^2 = \frac{-7}{-|a|} = \frac{7}{|a|}$ For real solutions, we need $\frac{7}{|a|} > 0$ , which is always true. But for the parabola to actually reach the x-axis from vertex height 7, we need the "stretch factor" $|a|$ to be large enough. The minimum occurs when the vertex-to-zero distance equals 7 units, giving us $|a| > \frac{7}{9}$ . Choice A gives the opposite inequality. Choice B requires $|a|$ to exceed 7, which is too restrictive. Choice D has the wrong inequality direction. Choice C correctly identifies that $|a| > \frac{7}{9}$ . Study tip: In vertex form problems involving zeros, always check whether the parabola orientation and vertex position allow x-intercepts to exist given the constraints.
Given that $f(x) = 3x^2 - 12x + 15$ can be written as $f(x) = 3(x - h)^2 + k$ , and $g(x) = x^2 - 4x + 5$ can be written as $g(x) = (x - m)^2 + n$ , what is the relationship between the vertices of these two functions?
Answer: The vertices have the same x-coordinate but different y-coordinates For f(x) = 3x² - 12x + 15: Factor out 3: f(x) = 3(x² - 4x + 5). Complete the square: x² - 4x + 5 = (x - 2)² - 4 + 5 = (x - 2)² + 1. So f(x) = 3[(x - 2)² + 1] = 3(x - 2)² + 3. The vertex is (2, 3). For g(x) = x² - 4x + 5: Complete the square: x² - 4x + 5 = (x - 2)² - 4 + 5 = (x - 2)² + 1. The vertex is (2, 1). Both vertices have x-coordinate 2, but different y-coordinates (3 and 1). Choice A is incorrect because the y-coordinates differ. Choice C reverses the relationship. Choice D incorrectly states the vertical distance as 3 units when it's actually 2 units.
For what value of $c$ will the expression $x^2 - 8x + c$ have exactly one real zero when factored?
Answer: $16$ When you encounter a quadratic expression that must have "exactly one real zero," you're dealing with a perfect square trinomial. This happens when the discriminant equals zero. For a quadratic $ax^2 + bx + c$ , the discriminant is $b^2 - 4ac$ . When the discriminant equals zero, the quadratic has exactly one real solution (a repeated root), meaning it factors as $(x - r)^2$ for some value $r$ . In $x^2 - 8x + c$ , we have $a = 1$ , $b = -8$ , and we need to find $c$ . Setting the discriminant to zero: $(-8)^2 - 4(1)(c) = 0$ , which gives us $64 - 4c = 0$ . Solving: $4c = 64$ , so $c = 16$ . We can verify this: $x^2 - 8x + 16 = (x - 4)^2$ , which has exactly one zero at $x = 4$ . Looking at the wrong answers: Choice A ( $c = 8$ ) gives $x^2 - 8x + 8$ , which has discriminant $64 - 32 = 32 > 0$ , producing two distinct real zeros. Choice C ( $c = 4$ ) gives $x^2 - 8x + 4$ , with discriminant $64 - 16 = 48 > 0$ , also yielding two real zeros. Choice D ( $c = 64$ ) gives $x^2 - 8x + 64$ , with discriminant $64 - 256 = -192 < 0$ , resulting in no real zeros. The answer is B. Study tip: For perfect square trinomials of the form $x^2 + bx + c$ , the constant term always equals $\left(\frac{b}{2}\right)^2$ . Here, $\left(\frac{-8}{2}\right)^2 = 16$ .
The quadratic $p(x) = x^2 + bx + 36$ has zeros at $x = -4$ and $x = -9$ . What is the value of $b$ ?
Answer: $13$ When you encounter a quadratic with known zeros, you're working with the relationship between a polynomial's factors and its expanded form. Since the zeros are the x-values where the function equals zero, you can use them to reconstruct the original quadratic. If $p(x) = x^2 + bx + 36$ has zeros at $x = -4$ and $x = -9$ , then the factored form is $p(x) = (x - (-4))(x - (-9)) = (x + 4)(x + 9)$ . Let's expand this: $(x + 4)(x + 9) = x^2 + 9x + 4x + 36 = x^2 + 13x + 36$ . Comparing this to the given form $x^2 + bx + 36$ , we see that $b = 13$ . You can also verify this using Vieta's formulas: for a quadratic $x^2 + bx + c$ , the sum of the roots equals $-b$ . Here, $(-4) + (-9) = -13$ , so $-b = -13$ , giving us $b = 13$ . Looking at the wrong answers: Choice A gives $b = -5$ , which would make the sum of roots $5$ , not $-13$ . Choice B gives $b = -13$ , which is the negative of what we need—this is the sum of the roots, not the coefficient. Choice C gives $b = 5$ , which would make the sum of roots $-5$ , again incorrect. Remember: for any quadratic $x^2 + bx + c$ , the sum of the roots always equals $-b$ . This relationship, combined with factoring, gives you two reliable paths to the same answer.
The quadratic $g(x) = 4x^2 - 16x + 12$ can be factored to reveal its zeros. Which statement about these zeros is correct?
Answer: Both zeros are positive integers and their product is 3 Factoring: g(x) = 4x² - 16x + 12 = 4(x² - 4x + 3) = 4(x - 1)(x - 3). The zeros are x = 1 and x = 3, both positive integers with product 1 × 3 = 3. Choice A: their sum is 4, but this describes the coefficient relationship, not a special property. Choice C incorrectly states one is negative. Choice D incorrectly calls them irrational.
A quadratic expression $2x^2 + 8x + k$ can be written as $2(x + 2)^2 + m$ for some constants $k$ and $m$ . What is the relationship between $k$ and $m$ ?
Answer: $k = m + 8$ This question tests your understanding of completing the square and vertex form transformations. When you see a quadratic that can be rewritten in vertex form, you're looking at two equivalent expressions that must be equal for all values of x. To find the relationship between k and m, expand the vertex form and compare coefficients. Starting with $2(x + 2)^2 + m$ , first expand the squared term: $(x + 2)^2 = x^2 + 4x + 4$ . Then multiply by 2: $2(x^2 + 4x + 4) = 2x^2 + 8x + 8$ . Finally, add m: $2x^2 + 8x + 8 + m$ . Since this must equal $2x^2 + 8x + k$ , the constant terms must be equal: $k = 8 + m$ , or $k = m + 8$ . Looking at the wrong answers: Choice A ( $k = 2m + 8$ ) incorrectly doubles m, likely from confusion about the coefficient 2 in front of the squared term. Choice B ( $k = m - 8$ ) has the wrong sign, suggesting a mistake when expanding $(x + 2)^2$ or combining the constant terms. Choice D ( $k = 2m - 8$ ) combines both errors—doubling m and using the wrong sign. Study tip: When working with vertex form transformations, always expand completely and compare coefficients systematically. The coefficient outside the squared term affects the x-terms when expanded, but the constant term relationship is usually more straightforward than it initially appears.
The expression $6x^2 - 24x + 18$ can be rewritten in the form $a(x - h)^2 + k$ . What is the minimum value of this expression?
Answer: $-6$ When you see a quadratic expression that needs to be rewritten in vertex form $a(x - h)^2 + k$ , you're being asked to complete the square. The key insight is that $k$ represents the minimum value of the expression (since the squared term is always non-negative). To complete the square for $6x^2 - 24x + 18$ , first factor out the coefficient of $x^2$ from the first two terms: $6(x^2 - 4x) + 18$ . Next, complete the square inside the parentheses. Take half of the coefficient of $x$ (which is $-4$ ), square it: $(-2)^2 = 4$ . Add and subtract this inside the parentheses: $6(x^2 - 4x + 4 - 4) + 18 = 6((x - 2)^2 - 4) + 18$ . Distribute and simplify: $6(x - 2)^2 - 24 + 18 = 6(x - 2)^2 - 6$ . Since $(x - 2)^2 \geq 0$ for all real $x$ , the minimum value occurs when $(x - 2)^2 = 0$ , giving us $6(0) - 6 = -6$ . Choice A ( $2$ ) is the value of $h$ in the vertex form, not the minimum value. Choice B ( $-24$ ) comes from incorrectly using $6 \times (-4) = -24$ without completing the square properly. Choice C ( $18$ ) is the constant term from the original expression, which students might mistakenly think is the minimum. Remember: in vertex form $a(x - h)^2 + k$ , the value $k$ is always the minimum (when $a > 0$ ) or maximum (when $a < 0$ ) of the quadratic function.
A quadratic function has the form $f(x) = ax^2 + bx + c$ where $a > 0$ . When rewritten in the form $f(x) = a(x - h)^2 + k$ , the vertex is at $(3, -8)$ . If one zero of the function is at $x = 1$ , what is the other zero?
Answer: $x = 5$ Since the vertex is at (3, -8), the axis of symmetry is x = 3. For a quadratic function, the zeros are equidistant from the axis of symmetry. If one zero is at x = 1, which is 2 units to the left of x = 3, then the other zero must be 2 units to the right of x = 3, at x = 5. Choice B uses the distance incorrectly as 4 units. Choice C places the second zero on the same side of the axis of symmetry. Choice D adds 3 to the given zero instead of using symmetry.
Consider the quadratic expression $4x^2 - 16x + 15$ . When rewritten in vertex form, the expression becomes $4(x - p)^2 + q$ . If this expression is then factored completely over the real numbers, what is the distance between its two zeros?
Answer: $1$ When you encounter a quadratic expression and need to find the distance between its zeros, you're working with concepts that connect vertex form, factoring, and the relationship between roots. Let's start by converting $4x^2 - 16x + 15$ to vertex form. Factor out the coefficient of $x^2$ : $4(x^2 - 4x) + 15$ . To complete the square inside the parentheses, take half of the coefficient of $x$ and square it: $(-4/2)^2 = 4$ . Add and subtract this inside: $4(x^2 - 4x + 4 - 4) + 15 = 4((x-2)^2 - 4) + 15 = 4(x-2)^2 - 16 + 15 = 4(x-2)^2 - 1$ . Now we can find the zeros by setting the expression equal to zero: $4(x-2)^2 - 1 = 0$ . Solving: $4(x-2)^2 = 1$ , so $(x-2)^2 = \frac{1}{4}$ , which gives us $x-2 = \pm\frac{1}{2}$ . Therefore, $x = 2 \pm \frac{1}{2}$ , so the zeros are $x = \frac{5}{2}$ and $x = \frac{3}{2}$ . The distance between the zeros is $\frac{5}{2} - \frac{3}{2} = 1$ , making C correct. Choice A ( $\frac{1}{2}$ ) represents half the distance between zeros. Choice B ( $2$ ) is the x-coordinate of the vertex, not the distance between roots. Choice D ( $\frac{3}{2}$ ) is one of the actual zeros, not the distance between them. Remember: the distance between quadratic roots equals the absolute difference of the two x-values where the expression equals zero.
A quadratic function $f(x) = ax^2 + bx + c$ has the property that $f(2) = 0$ and $f(-3) = 0$ . If the function can be written in the form $f(x) = a(x - h)^2 + k$ , what is the value of $h$ ?
Answer: $-0.5$ Since f(2) = 0 and f(-3) = 0, the zeros are x = 2 and x = -3. The vertex form f(x) = a(x - h)² + k has h-coordinate at the midpoint of the zeros: h = (2 + (-3))/2 = -1/2 = -0.5. Choice B uses the positive value. Choice C incorrectly uses the distance between zeros. Choice D uses the negative of the distance.
The expression $(2x - 6)(x + 4)$ reveals information about a quadratic function. What is the y-intercept of this function?
Answer: $-24$ When you see a quadratic function in factored form like $(2x - 6)(x + 4)$ , you can find the y-intercept by substituting $x = 0$ into the expression. The y-intercept occurs where the graph crosses the y-axis, which is always at the point where $x = 0$ . To find the y-intercept, substitute $x = 0$ into $(2x - 6)(x + 4)$ : $(2(0) - 6)(0 + 4) = (-6)(4) = -24$ So the y-intercept is $-24$ , which is answer choice D. Let's examine why the other answers are incorrect. Answer choice A ( $24$ ) represents a sign error — you might get this if you calculated $(-6)(4)$ as positive instead of negative. Answer choice B ( $6$ ) comes from misunderstanding what the y-intercept means; this is the constant term in the first factor, but it's not the y-intercept. Answer choice C ( $-4$ ) is the constant term from the second factor, which also doesn't give you the y-intercept. Remember this key strategy: to find the y-intercept of any function, always substitute $x = 0$ . Whether the quadratic is in factored form, standard form, or vertex form, this method works consistently. Don't get distracted by the individual numbers within the factors — you need to evaluate the entire expression at $x = 0$ to find where the parabola crosses the y-axis.
The expression $3x^2 - 12x + 15$ can be rewritten to reveal its minimum value. What is this minimum value?
Answer: $3$ Completing the square: 3x² - 12x + 15 = 3(x² - 4x) + 15 = 3(x² - 4x + 4 - 4) + 15 = 3((x - 2)² - 4) + 15 = 3(x - 2)² - 12 + 15 = 3(x - 2)² + 3. The minimum value is 3. Choice B uses the coefficient of x. Choice C uses the constant term. Choice D results from forgetting to distribute the 3 when completing the square.
A parabola has equation $y = x^2 + 4x - 5$ . To find where this parabola intersects the line $y = 7$ , which form of the quadratic equation should be solved?
Answer: $x^2 + 4x - 12 = 0$ When finding where a parabola intersects a horizontal line, you need to set the parabola's equation equal to the line's y-value and solve for x. This creates a quadratic equation that tells you the x-coordinates of intersection points. To find where $y = x^2 + 4x - 5$ intersects $y = 7$ , you set these equal: $x^2 + 4x - 5 = 7$ . Moving everything to one side gives you $x^2 + 4x - 5 - 7 = 0$ , which simplifies to $x^2 + 4x - 12 = 0$ . This matches answer choice B. Answer A shows $(x + 2)^2 - 9 = 7$ . While this comes from the correct vertex form of the original parabola, it hasn't been simplified to standard form. You'd still need to expand and rearrange this to solve it efficiently. Answer C gives $(x + 5)(x - 1) = 7$ . This represents the factored form of the original parabola (since expanding gives $x^2 + 4x - 5$ ), but it incorrectly sets this equal to 7 without rearranging. To solve this, you'd need to subtract 7 first. Answer D shows $x^2 + 4x + 5 = 7$ . This contains a sign error in the constant term—it should be $-5$ , not $+5$ . Remember: when finding intersections between a parabola and a horizontal line, always set the expressions equal, then rearrange into standard form ( $ax^2 + bx + c = 0$ ) before solving. This avoids algebraic errors and makes factoring or using the quadratic formula straightforward.
The quadratic $f(x) = -2x^2 + 12x - 10$ can be rewritten to show that its maximum occurs at $x = 3$ . What is the maximum value?
Answer: $8$ Completing the square: f(x) = -2x² + 12x - 10 = -2(x² - 6x) - 10 = -2(x² - 6x + 9 - 9) - 10 = -2((x - 3)² - 9) - 10 = -2(x - 3)² + 18 - 10 = -2(x - 3)² + 8. The maximum value is 8. Choice B forgets to subtract 10. Choice C uses 2(3² - 6(3)) + 18. Choice D adds instead of subtracting when distributing -2.
Which factored form of $2x^2 + 8x - 42$ most clearly reveals that the parabola crosses the x-axis at $x = -7$ and $x = 3$ ?
Answer: $2(x + 7)(x - 3)$ To reveal zeros at x = -7 and x = 3, we need factors (x + 7) and (x - 3). Factoring: 2x² + 8x - 42 = 2(x² + 4x - 21) = 2(x + 7)(x - 3). Choice B reverses the signs inside parentheses. Choices C and D don't factor out 2 completely, making the zeros less obvious.
A quadratic function $h(x) = 3(x - 1)^2 - 12$ is given in vertex form. To find the x-intercepts algebraically, what equation should be solved first?
Answer: $(x - 1)^2 = 4$ When you need to find x-intercepts of any function, you're looking for the points where the graph crosses the x-axis. This happens when the function's output equals zero, so you set the function equal to zero and solve for x. Starting with $h(x) = 3(x - 1)^2 - 12$ , you set this equal to zero: $3(x - 1)^2 - 12 = 0$ . To solve this equation, you need to isolate the squared term. Add 12 to both sides: $3(x - 1)^2 = 12$ . Then divide both sides by 3: $(x - 1)^2 = 4$ . This matches answer choice D. Choice A, $x^2 - 2x + 1 = 4$ , represents what you'd get if you expanded the vertex form and then tried to solve, but this approach skips the crucial step of setting the function equal to zero first. Choice B, $3(x - 1)^2 = 12$ , is actually a correct intermediate step in the solving process, but it's not the equation you should solve "first" - you still need to divide by 3 to isolate the squared term. Choice C, $(x - 1)^2 = -12$ , would result from incorrectly subtracting 12 instead of adding it when moving the constant term. Remember that finding x-intercepts always starts with setting the function equal to zero. In vertex form problems, work systematically: set equal to zero, isolate the squared term completely, then take the square root of both sides to finish solving.
A quadratic expression $x^2 + 6x + k$ can be written as $(x + p)^2 + q$ where $q = -5$ . What is the value of $k$ ?
Answer: $4$ Completing the square for x² + 6x + k: x² + 6x + k = (x + 3)² - 9 + k = (x + 3)² + (k - 9). Since this equals (x + p)² + q with q = -5, we have k - 9 = -5, so k = 4. Choice B is 3², the square of half the coefficient. Choice C adds 9 to -5. Choice D uses q directly.
The quadratic function $g(x) = -3x^2 + 18x - 24$ can be rewritten to reveal its vertex. In vertex form $g(x) = a(x - h)^2 + k$ , what is the value of $k$ ?
Answer: $3$ This question tests your ability to convert a quadratic function from standard form to vertex form by completing the square. The vertex form $g(x) = a(x - h)^2 + k$ immediately reveals the vertex at point $(h, k)$ , where $k$ represents the y-coordinate of the vertex. To find $k$ , you need to complete the square. Starting with $g(x) = -3x^2 + 18x - 24$ , first factor out the coefficient of $x^2$ from the first two terms: $g(x) = -3(x^2 - 6x) - 24$ . To complete the square inside the parentheses, take half of the coefficient of $x$ (which is $-6$ ), square it: $(-6/2)^2 = 9$ . Add and subtract this inside the parentheses: $g(x) = -3(x^2 - 6x + 9 - 9) - 24 = -3((x - 3)^2 - 9) - 24$ . Distributing the $-3$ : $g(x) = -3(x - 3)^2 + 27 - 24 = -3(x - 3)^2 + 3$ . Therefore, $k = 3$ . Choice A ( $-3$ ) is the value of $a$ , the coefficient that determines the parabola's direction and width. Choice B ( $6$ ) might come from mishandling the coefficient when completing the square. Choice D ( $9$ ) is the perfect square term you add and subtract during the completion process, but it gets multiplied by $-3$ and combined with the constant term. Remember: when completing the square, carefully track how factored coefficients affect the perfect square term when you distribute back through the expression.
The quadratic $Q(x) = 3x^2 - 18x + 24$ can be rewritten to reveal that its vertex has coordinates $(h, k)$ . What is the value of $h + k$ ?
Answer: $0$ When you encounter a quadratic function and need to find its vertex, you're working with the relationship between standard form and vertex form. The vertex form of a quadratic is $Q(x) = a(x - h)^2 + k$ , where $(h, k)$ represents the vertex coordinates. To find the vertex of $Q(x) = 3x^2 - 18x + 24$ , complete the square. First, factor out the coefficient of $x^2$ from the first two terms: $Q(x) = 3(x^2 - 6x) + 24$ . To complete the square inside the parentheses, take half of the coefficient of $x$ and square it: $(-6 \div 2)^2 = 9$ . Add and subtract this inside the parentheses: $Q(x) = 3(x^2 - 6x + 9 - 9) + 24 = 3((x - 3)^2 - 9) + 24$ . Distribute and simplify: $Q(x) = 3(x - 3)^2 - 27 + 24 = 3(x - 3)^2 - 3$ . From this vertex form, you can see that $h = 3$ and $k = -3$ , so $h + k = 3 + (-3) = 0$ . Choice A) $6$ likely comes from adding the coefficients incorrectly or confusing $h$ with $2h$ . Choice B) $3$ represents just the $h$ -coordinate, forgetting to include $k$ . Choice D) $-3$ represents just the $k$ -coordinate, ignoring $h$ . Remember: when completing the square, carefully track your arithmetic, especially when distributing coefficients. The vertex form immediately reveals both coordinates, making $h + k$ calculations straightforward.

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