Math 2 Quiz: Factored Form And Zeros
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Factored Form And ZerosQuestion 1 of 20

The profit function for a small business is P(x)=−x2+12x−35P(x) = -x^2 + 12x - 35 dollars, where xx is the number of items sold. At what production levels does the business break even (profit equals zero)?

x=5x = 5 and x=7x = 7 items
x=−5x = -5 and x=−7x = -7 items
x=6x = 6 items only (double root)
The business never breaks even (no real solutions)
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Math 2 Quiz

Math 2 Quiz: Factored Form And Zeros

Practice Factored Form And Zeros in Math 2 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 Factored Form And Zeros, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 2.

How to use this quiz

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

The profit function for a small business is P(x)=−x2+12x−35P(x) = -x^2 + 12x - 35 dollars, where xx is the number of items sold. At what production levels does the business break even (profit equals zero)?

  1. x=5x = 5 and x=7x = 7 items (correct answer)
  2. x=−5x = -5 and x=−7x = -7 items
  3. x=6x = 6 items only (double root)
  4. The business never breaks even (no real solutions)
Explanation: Set P(x)=0P(x) = 0: −x2+12x−35=0-x^2 + 12x - 35 = 0, or x2−12x+35=0x^2 - 12x + 35 = 0. Factoring: (x−5)(x−7)=0(x - 5)(x - 7) = 0, so x=5x = 5 or x=7x = 7. The business breaks even when producing 5 or 7 items. Choice B gives negative values which are meaningless in context. Choice C results from incorrectly finding the vertex instead of zeros. Choice D comes from computational errors in factoring.

Question 2

The profit function for a business is P(x)=−x2+8x−15P(x) = -x^2 + 8x - 15 thousand dollars, where xx is the number of products sold (in hundreds). To find the break-even points, the equation P(x)=0P(x) = 0 must be solved. What does the factored form reveal?

  1. P(x)=−(x−3)(x−5)P(x) = -(x - 3)(x - 5); break-even occurs at 300 and 500 products with maximum profit between (correct answer)
  2. P(x)=−(x+3)(x+5)P(x) = -(x + 3)(x + 5); break-even occurs at 300 and 500 products with maximum profit between
  3. P(x)=−(x−3)(x−5)P(x) = -(x - 3)(x - 5); break-even occurs at 3 and 5 products with maximum loss between
  4. P(x)=(x−3)(x−5)P(x) = (x - 3)(x - 5); break-even occurs at 300 and 500 products with minimum profit between
Explanation: Factoring: P(x)=−(x2−8x+15)=−(x−3)(x−5)P(x) = -(x^2 - 8x + 15) = -(x - 3)(x - 5). The zeros are x=3x = 3 and x=5x = 5. Since xx represents hundreds of products, break-even occurs at 300 and 500 products. Since the coefficient of x2x^2 is negative, the parabola opens downward, meaning profit is positive (maximum) between the break-even points. Choice B has wrong signs in factors. Choice C misinterprets the profit between break-even points. Choice D has the wrong leading coefficient sign.

Question 3

Consider the quadratic f(x)=4x2−16x+12f(x) = 4x^2 - 16x + 12. A student claims the factored form is f(x)=4(x−1)(x−3)f(x) = 4(x - 1)(x - 3) and concludes the xx-intercepts are at x=4x = 4 and x=16x = 16. Which part of their reasoning contains an error?

  1. The factored form is incorrect; it should be f(x)=4(x−2)(x−6)f(x) = 4(x - 2)(x - 6) with intercepts at x=2x = 2 and x=6x = 6
  2. The factored form is correct, but the xx-intercepts should be at x=1x = 1 and x=3x = 3, not x=4x = 4 and x=16x = 16 (correct answer)
  3. The factored form is incorrect; it should be f(x)=(x−4)(x−12)f(x) = (x - 4)(x - 12) with intercepts at x=4x = 4 and x=12x = 12
  4. Both the factored form and intercepts are incorrect; the correct form is f(x)=4(x+1)(x+3)f(x) = 4(x + 1)(x + 3) with intercepts at x=−1x = -1 and x=−3x = -3
Explanation: Let's verify the factored form: f(x)=4x2−16x+12=4(x2−4x+3)=4(x−1)(x−3)f(x) = 4x^2 - 16x + 12 = 4(x^2 - 4x + 3) = 4(x-1)(x-3). Check: (x−1)(x−3)=x2−4x+3(x-1)(x-3) = x^2 - 4x + 3 ✓. So the factored form is correct. However, the xx-intercepts occur where f(x)=0f(x) = 0, which means 4(x−1)(x−3)=04(x-1)(x-3) = 0, giving x=1x = 1 and x=3x = 3. The student confused the factors with something else. Choice A has wrong factorization. Choice C ignores the factor of 4. Choice D has wrong signs throughout.

Question 4

Given f(x)=x2−2x−15f(x) = x^2 - 2x - 15, a student factors this as f(x)=(x−5)(x+3)f(x) = (x - 5)(x + 3) and then states that the parabola crosses the xx-axis at exactly three points: x=5x = 5, x=−3x = -3, and x=0x = 0. What error did the student make?

  1. The factorization is wrong; it should be (x+5)(x−3)(x + 5)(x - 3) giving different xx-intercepts entirely
  2. The factorization is correct, but x=0x = 0 is not an xx-intercept; only x=5x = 5 and x=−3x = -3 are intercepts (correct answer)
  3. The factorization is correct, but parabolas can only have at most two xx-intercepts, so three is impossible
  4. Both the factorization and the claim about three intercepts are wrong; the correct intercepts are x=−5x = -5 and x=3x = 3
Explanation: The factorization f(x)=(x−5)(x+3)f(x) = (x - 5)(x + 3) is correct: (x−5)(x+3)=x2+3x−5x−15=x2−2x−15(x-5)(x+3) = x^2 + 3x - 5x - 15 = x^2 - 2x - 15 ✓. The zeros occur where (x−5)(x+3)=0(x-5)(x+3) = 0, giving x=5x = 5 and x=−3x = -3. To check if x=0x = 0 is an intercept: f(0)=02−2(0)−15=−15≠0f(0) = 0^2 - 2(0) - 15 = -15 \neq 0. So x=0x = 0 is not an xx-intercept. Choice A gives incorrect factorization. Choice C is correct about the maximum but doesn't identify the specific error. Choice D is completely wrong about both aspects.

Question 5

For the function g(x)=x2−6x+9g(x) = x^2 - 6x + 9, a student writes the factored form as g(x)=(x−3)2g(x) = (x - 3)^2 and claims there are two xx-intercepts at x=3x = 3 and x=3x = 3. How should this reasoning be corrected?

  1. The factored form is wrong; it should be g(x)=(x−3)(x+3)g(x) = (x - 3)(x + 3) giving intercepts at x=3x = 3 and x=−3x = -3
  2. The factored form is correct, but there is only one xx-intercept at x=3x = 3 with multiplicity 2 (correct answer)
  3. The factored form is correct, and there are indeed two distinct xx-intercepts, both located at x=3x = 3
  4. The factored form is wrong; it should be g(x)=(x+3)2g(x) = (x + 3)^2 giving one intercept at x=−3x = -3 with multiplicity 2
Explanation: The factored form g(x)=(x−3)2g(x) = (x-3)^2 is correct since x2−6x+9=(x−3)2x^2 - 6x + 9 = (x-3)^2. However, this represents a repeated root or double root at x=3x = 3, not two separate intercepts. There is one xx-intercept at x=3x = 3 where the parabola touches but doesn't cross the xx-axis. Choice A gives wrong factorization. Choice C misunderstands what a double root means graphically. Choice D has incorrect signs in the factorization.

Question 6

The function f(x)=2x2−18x+40f(x) = 2x^2 - 18x + 40 represents the cost in dollars for producing xx items. To find when the cost equals zero (break-even), the function is factored. What are the xx-intercepts and their meaning?

  1. x=4x = 4 and x=5x = 5; the company breaks even when producing 4 or 5 items (correct answer)
  2. x=−4x = -4 and x=−5x = -5; the company breaks even when producing 4 or 5 items
  3. x=4x = 4 and x=5x = 5; the company has minimum cost when producing 4 or 5 items
  4. x=2x = 2 and x=10x = 10; the company breaks even when producing 2 or 10 items
Explanation: Factor out 2: f(x)=2(x2−9x+20)=2(x−4)(x−5)f(x) = 2(x^2 - 9x + 20) = 2(x - 4)(x - 5). The zeros are x=4x = 4 and x=5x = 5, meaning the cost equals zero when producing 4 or 5 items (break-even points). Choice B has wrong signs for the zeros but correct interpretation. Choice C has correct zeros but confuses break-even with minimum cost. Choice D has incorrect zeros from factoring errors.

Question 7

The area of a rectangular garden is given by A(w)=w2+12w+32A(w) = w^2 + 12w + 32 square feet, where ww is the width in feet. If the area formula can be factored as A(w)=(w+a)(w+b)A(w) = (w + a)(w + b), what do the values −a-a and −b-b represent in this context?

  1. The values −a=−4-a = -4 and −b=−8-b = -8 represent impossible negative dimensions for the garden width
  2. The values −a=−4-a = -4 and −b=−8-b = -8 represent the width and length when the garden area is minimized
  3. The values −a=−8-a = -8 and −b=−4-b = -4 represent impossible negative dimensions for the garden width
  4. The values −a=−4-a = -4 and −b=−8-b = -8 represent the xx-intercepts where the area equals zero square feet (correct answer)
Explanation: Factoring w2+12w+32=(w+4)(w+8)w^2 + 12w + 32 = (w + 4)(w + 8), so a=4a = 4 and b=8b = 8. The values −a=−4-a = -4 and −b=−8-b = -8 are the ww-values where A(w)=0A(w) = 0, meaning the area equals zero. While these are negative and thus not physically meaningful for width, they are mathematically the xx-intercepts of the function. Choice A correctly identifies the values but focuses on physical impossibility rather than mathematical meaning. Choices B and C misinterpret what these values represent mathematically.

Question 8

A manufacturing company's daily profit is modeled by the function P(x)=−x2+14x−48P(x) = -x^2 + 14x - 48, where xx represents the number of items produced (in hundreds) and P(x)P(x) represents the profit in thousands of dollars.

The company wants to determine the production levels where they break even (P(x)=0P(x) = 0). After factoring, what do the results indicate about their production strategy?

  1. Break-even at 600 and 800 items; unprofitable production range is between these values with minimum loss at 700 items
  2. Break-even at 6 and 8 items; profitable production range is between these values with maximum profit at 7 items
  3. Break-even at 600 and 800 items; profitable production range is between these values with maximum profit at 700 items (correct answer)
  4. Break-even at 6 and 8 items; production should exceed 800 items to maximize profit since the parabola opens upward
Explanation: When you encounter a quadratic profit function, you're looking at a parabola that models how profit changes with production. The key is to find the break-even points (where profit equals zero) and understand the parabola's shape to determine optimal production strategy. To find break-even points, set P(x)=−x2+14x−48=0P(x) = -x^2 + 14x - 48 = 0. Factoring this quadratic: −(x2−14x+48)=0-(x^2 - 14x + 48) = 0, which gives us −(x−6)(x−8)=0-(x - 6)(x - 8) = 0. Therefore, x=6x = 6 or x=8x = 8. Since xx represents hundreds of items, the break-even points are at 600 and 800 items. Since the coefficient of x2x^2 is negative (-1), this parabola opens downward, meaning profit is positive between the break-even points and negative outside them. The maximum profit occurs at the vertex, halfway between the roots: x=6+82=7x = \frac{6 + 8}{2} = 7 (or 700 items). Answer A incorrectly states the company is unprofitable between break-even points with minimum loss at 700 items—this confuses the parabola's orientation. Answer B correctly identifies break-even at 6 and 8 items but fails to convert to actual production numbers (600 and 800). Answer D makes two critical errors: suggesting production should exceed 800 items for maximum profit and claiming the parabola opens upward. Remember: when working with quadratic models, always check the leading coefficient to determine whether the parabola opens up or down, as this tells you whether the vertex represents a maximum or minimum value.

Question 9

The height of a ball thrown upward is modeled by h(t)=−16t2+48t+64h(t) = -16t^2 + 48t + 64. If this function factors as h(t)=−16(t+1)(t−4)h(t) = -16(t + 1)(t - 4), what is the physical significance of each zero in the context of the problem?

  1. t=−1t = -1 represents the ball being 1 second below ground; t=4t = 4 represents when the ball hits the ground
  2. t=−1t = -1 represents a mathematical artifact with no physical meaning; t=4t = 4 represents when the ball hits the ground (correct answer)
  3. t=−1t = -1 represents when the ball was thrown; t=4t = 4 represents the maximum height time
  4. Both t=−1t = -1 and t=4t = 4 represent times when the ball is at the same height above ground
Explanation: The zeros t=−1t = -1 and t=4t = 4 both represent times when h(t)=0h(t) = 0 (height = 0). However, t=−1t = -1 represents a time before the ball was thrown (negative time), which has no physical meaning in this context. Only t=4t = 4 represents a physically meaningful time when the ball hits the ground. Choice A incorrectly interprets negative time. Choice C misidentifies what the zeros represent. Choice D is mathematically correct but ignores the physical context.

Question 10

A rectangular garden's area in square feet is given by A(w)=w2+7w−18A(w) = w^2 + 7w - 18, where ww is the width in feet. For what width(s) would the garden have zero area, and what does this mean in the context?

  1. w=2w = 2 and w=−9w = -9; only w=2w = 2 is meaningful (garden collapses to a line)
  2. w=−2w = -2 and w=9w = 9; only w=9w = 9 is meaningful (garden collapses to a line) (correct answer)
  3. w=3w = 3 and w=−6w = -6; both values represent different garden orientations
  4. w=−3w = -3 and w=6w = 6; both values represent different garden orientations
Explanation: Set A(w)=0A(w) = 0: w2+7w−18=0w^2 + 7w - 18 = 0. Factoring: (w−9)(w+2)=0(w - 9)(w + 2) = 0, so w=9w = 9 or w=−2w = -2. Since width cannot be negative, only w=9w = 9 feet is meaningful. When area equals zero, the garden has collapsed to a line (zero width or length). Choice A has incorrect factors. Choice C and D have wrong factorizations and incorrectly suggest both solutions are meaningful.

Question 11

A quadratic function intersects the x-axis at two points. When written in factored form, one factor is (x−4)(x - 4) and the leading coefficient is −12-\frac{1}{2}.

If the axis of symmetry of this parabola is at x=1x = 1, what is the complete factored form of the quadratic?

  1. −12(x−4)(x+1)-\frac{1}{2}(x - 4)(x + 1)
  2. −12(x−4)(x−2)-\frac{1}{2}(x - 4)(x - 2)
  3. −12(x−4)(x+2)-\frac{1}{2}(x - 4)(x + 2) (correct answer)
  4. −12(x−4)(x−6)-\frac{1}{2}(x - 4)(x - 6)
Explanation: When you encounter a quadratic function with given factors and symmetry properties, you need to use the relationship between roots and the axis of symmetry to find the missing information. Since the quadratic intersects the x-axis at two points, these are the roots of the function. You're told one factor is (x−4)(x - 4), which means x=4x = 4 is one root. The axis of symmetry for any parabola lies exactly halfway between its two roots. If the axis of symmetry is at x=1x = 1 and one root is at x=4x = 4, you can find the other root using the midpoint formula. The axis of symmetry x=1x = 1 equals root1+root22\frac{\text{root}_1 + \text{root}_2}{2}. Substituting: 1=4+root221 = \frac{4 + \text{root}_2}{2}. Solving: 2=4+root22 = 4 + \text{root}_2, so root2=−2\text{root}_2 = -2. This means the second factor is (x−(−2))=(x+2)(x - (-2)) = (x + 2). With the leading coefficient −12-\frac{1}{2}, the complete factored form is −12(x−4)(x+2)-\frac{1}{2}(x - 4)(x + 2). Choice A gives roots at x=4x = 4 and x=−1x = -1, creating an axis of symmetry at x=1.5x = 1.5, not x=1x = 1. Choice B gives roots at x=4x = 4 and x=2x = 2, creating an axis of symmetry at x=3x = 3. Choice D gives roots at x=4x = 4 and x=6x = 6, creating an axis of symmetry at x=5x = 5. Remember: the axis of symmetry is always the average of the two roots. Use this relationship to quickly check whether your factored form matches the given symmetry condition.

Question 12

A student claims that 3x2−15x+183x^2 - 15x + 18 factors as 3(x−2)(x−3)3(x - 2)(x - 3). To verify this factorization, what should be checked, and is the student correct?

  1. Check by expanding: 3(x−2)(x−3)=3x2−9x+12≠3x2−15x+183(x - 2)(x - 3) = 3x^2 - 9x + 12 \neq 3x^2 - 15x + 18. The student is incorrect.
  2. Check by substituting x=2x = 2 and x=3x = 3: both give zero. The student is correct.
  3. Check by expanding: 3(x−2)(x−3)=3x2−15x+183(x - 2)(x - 3) = 3x^2 - 15x + 18. The student is correct. (correct answer)
  4. Check by finding zeros of original: x=5x = 5 and x=6x = 6. The student is incorrect.
Explanation: When verifying a polynomial factorization, the most reliable method is to expand the proposed factorization and check if it equals the original polynomial. This ensures you've correctly identified all terms and coefficients. Let's expand 3(x−2)(x−3)3(x - 2)(x - 3) step by step. First, multiply the binomials: (x−2)(x−3)=x2−3x−2x+6=x2−5x+6(x - 2)(x - 3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6. Then multiply by 3: 3(x2−5x+6)=3x2−15x+183(x^2 - 5x + 6) = 3x^2 - 15x + 18. This exactly matches the original polynomial, confirming the student is correct. Answer A makes an expansion error. When multiplying (x−2)(x−3)(x - 2)(x - 3), they likely forgot one of the middle terms or made an arithmetic mistake, getting 3x2−9x+123x^2 - 9x + 12 instead of the correct 3x2−15x+183x^2 - 15x + 18. Answer B uses zero-checking, which tests whether x=2x = 2 and x=3x = 3 are roots. While this method can verify that these values are zeros of the original polynomial, it's incomplete for factorization verification. A polynomial could have the correct zeros but an incorrect leading coefficient or additional factors. Answer D incorrectly identifies the zeros. To find the actual zeros of 3x2−15x+18=03x^2 - 15x + 18 = 0, divide by 3 to get x2−5x+6=0x^2 - 5x + 6 = 0, which factors as (x−2)(x−3)=0(x - 2)(x - 3) = 0, giving zeros at x=2x = 2 and x=3x = 3, not 5 and 6. Always verify factorizations by expanding back to the original form—it catches both coefficient errors and missing factors that zero-checking might miss.

Question 13

A projectile's height h(t)=−16t2+64t+80h(t) = -16t^2 + 64t + 80 feet above ground after tt seconds. If the projectile hits the ground when h(t)=0h(t) = 0, which statement about the factored form is correct?

  1. The factored form is −16(t+1)(t−5)-16(t + 1)(t - 5) and the projectile hits the ground at t=5t = 5 seconds (correct answer)
  2. The factored form is −16(t−1)(t+5)-16(t - 1)(t + 5) and the projectile hits the ground at t=1t = 1 second
  3. The factored form is −16(t+1)(t−5)-16(t + 1)(t - 5) and the projectile hits the ground at t=−1t = -1 seconds
  4. The factored form is 16(t+1)(t−5)16(t + 1)(t - 5) and the projectile hits the ground at t=5t = 5 seconds
Explanation: First factor out -16: h(t)=−16(t2−4t−5)h(t) = -16(t^2 - 4t - 5). Then factor the quadratic: t2−4t−5=(t+1)(t−5)t^2 - 4t - 5 = (t + 1)(t - 5), so h(t)=−16(t+1)(t−5)h(t) = -16(t + 1)(t - 5). The zeros are t=−1t = -1 and t=5t = 5. Since time cannot be negative in this context, the projectile hits the ground at t=5t = 5 seconds. Choice B has wrong signs in factors. Choice C correctly identifies the factored form but incorrectly chooses the negative time. Choice D has the wrong leading coefficient sign.

Question 14

The function h(x)=2x2+8x+6h(x) = 2x^2 + 8x + 6 represents the height of a structure. When factored completely, which statement correctly describes both the mathematical form and the meaning of any zeros?

  1. h(x)=2(x−1)(x−3)h(x) = 2(x - 1)(x - 3) with zeros at x=1x = 1 and x=3x = 3; the structure has zero height at these positions
  2. h(x)=(2x+2)(x+3)h(x) = (2x + 2)(x + 3) with zeros at x=−1x = -1 and x=−3x = -3; the structure height is zero at these positions
  3. h(x)=2(x+1)(x+3)h(x) = 2(x + 1)(x + 3) with zeros at x=−1x = -1 and x=−3x = -3; negative inputs may not be physically meaningful (correct answer)
  4. h(x)=2(x+1)(x+3)h(x) = 2(x + 1)(x + 3) with zeros at x=1x = 1 and x=3x = 3; these represent maximum height positions
Explanation: When you encounter a quadratic function that needs to be factored, start by looking for common factors, then use appropriate factoring techniques to find the complete factored form and zeros. To factor h(x)=2x2+8x+6h(x) = 2x^2 + 8x + 6, first factor out the common factor of 2: h(x)=2(x2+4x+3)h(x) = 2(x^2 + 4x + 3). Now factor the quadratic expression x2+4x+3x^2 + 4x + 3. You need two numbers that multiply to 3 and add to 4, which are 1 and 3. This gives you h(x)=2(x+1)(x+3)h(x) = 2(x + 1)(x + 3). To find the zeros, set each factor equal to zero: x+1=0x + 1 = 0 gives x=−1x = -1, and x+3=0x + 3 = 0 gives x=−3x = -3. Choice A incorrectly factors to 2(x−1)(x−3)2(x - 1)(x - 3) and claims zeros at positive values, but expanding this gives 2x2−8x+62x^2 - 8x + 6, which doesn't match the original function. Choice B has the factorization (2x+2)(x+3)(2x + 2)(x + 3), but this isn't completely factored since you can still factor out 2 from the first term. Choice D has the correct factored form but incorrectly states the zeros as x=1x = 1 and x=3x = 3, when they're actually x=−1x = -1 and x=−3x = -3. Choice C correctly identifies both the complete factorization and the actual zeros, plus appropriately notes that negative input values may not be physically meaningful when interpreting height in real-world contexts. Remember: always check your factoring by expanding back to the original form, and consider whether mathematical solutions make sense in the given context.

Question 15

A quadratic function has the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c where a<0a < 0. If the factored form is f(x)=−2(x+3)(x−7)f(x) = -2(x + 3)(x - 7), which statement about the xx-intercepts and the parabola is correct?

  1. The xx-intercepts are at x=−3x = -3 and x=7x = 7, and the parabola opens upward between these points
  2. The xx-intercepts are at x=3x = 3 and x=−7x = -7, and the parabola opens downward between these points
  3. The xx-intercepts are at x=−3x = -3 and x=7x = 7, and the parabola opens downward with vertex above the xx-axis (correct answer)
  4. The xx-intercepts are at x=−3x = -3 and x=7x = 7, and the parabola opens upward with vertex below the xx-axis
Explanation: From f(x)=−2(x+3)(x−7)f(x) = -2(x + 3)(x - 7), setting f(x)=0f(x) = 0 gives x=−3x = -3 and x=7x = 7 as the xx-intercepts. Since a=−2<0a = -2 < 0, the parabola opens downward. For a downward-opening parabola with two xx-intercepts, the vertex must be above the xx-axis (at the maximum point). Choice A has correct intercepts but wrong direction. Choice B has wrong intercepts. Choice D has correct intercepts but impossible geometry (upward opening with a<0a < 0).

Question 16

A quadratic function f(x)=ax2+bx+cf(x) = ax^2 + bx + c has factored form f(x)=3(x+2)(x−6)f(x) = 3(x + 2)(x - 6). If the graph passes through the point (0,k)(0, k), what can be concluded about the yy-intercept and zeros?

  1. The zeros are x=3x = 3 and x=−2x = -2; the yy-intercept is at (0,−36)(0, -36)
  2. The zeros are x=2x = 2 and x=−6x = -6; the yy-intercept is at (0,36)(0, 36)
  3. The zeros are x=−2x = -2 and x=6x = 6; the yy-intercept is at (0,36)(0, 36)
  4. The zeros are x=−2x = -2 and x=6x = 6; the yy-intercept is at (0,−36)(0, -36) (correct answer)
Explanation: When you encounter a quadratic in factored form, you can directly read the zeros and calculate the y-intercept by substituting specific values. For f(x)=3(x+2)(x−6)f(x) = 3(x + 2)(x - 6), the zeros occur when the function equals zero. This happens when either factor equals zero: x+2=0x + 2 = 0 or x−6=0x - 6 = 0. Solving these gives x=−2x = -2 and x=6x = 6. To find the y-intercept, substitute x=0x = 0 into the function: f(0)=3(0+2)(0−6)=3(2)(−6)=−36f(0) = 3(0 + 2)(0 - 6) = 3(2)(-6) = -36 Therefore, the y-intercept is at (0,−36)(0, -36). Looking at the wrong answers: Choice A incorrectly identifies the zeros as x=3x = 3 and x=−2x = -2, likely confusing the coefficient 3 with a zero. Choice B makes sign errors on both zeros (x=2x = 2 and x=−6x = -6) and gets a positive y-intercept, suggesting calculation mistakes. Choice C correctly identifies the zeros but incorrectly calculates the y-intercept as positive 36 instead of negative 36, missing the negative sign from the (0−6)(0 - 6) factor. When working with factored form a(x−r1)(x−r2)a(x - r_1)(x - r_2), remember that the zeros are the values that make each factor zero, so if you see (x+2)(x + 2), the zero is x=−2x = -2, not x=2x = 2. For the y-intercept, always substitute x=0x = 0 and carefully track positive and negative signs through your multiplication.

Question 17

A quadratic function has the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c and can be factored as f(x)=−2(x+3)(x−1)f(x) = -2(x + 3)(x - 1). What are the zeros of this function and what is the value of the y-intercept?

  1. Zeros at x=3,−1x = 3, -1; y-intercept at y=6y = 6
  2. Zeros at x=3,−1x = 3, -1; y-intercept at y=−6y = -6
  3. Zeros at x=−3,1x = -3, 1; y-intercept at y=−6y = -6
  4. Zeros at x=−3,1x = -3, 1; y-intercept at y=6y = 6 (correct answer)
Explanation: When you encounter a quadratic function in factored form, you can directly read the zeros and calculate the y-intercept using specific techniques. To find the zeros of f(x)=−2(x+3)(x−1)f(x) = -2(x + 3)(x - 1), set the function equal to zero: −2(x+3)(x−1)=0-2(x + 3)(x - 1) = 0. Since the coefficient −2-2 is never zero, you need (x+3)(x−1)=0(x + 3)(x - 1) = 0. This happens when either factor equals zero: x+3=0x + 3 = 0 gives x=−3x = -3, and x−1=0x - 1 = 0 gives x=1x = 1. So the zeros are at x=−3x = -3 and x=1x = 1. For the y-intercept, substitute x=0x = 0 into the function: f(0)=−2(0+3)(0−1)=−2(3)(−1)=6f(0) = -2(0 + 3)(0 - 1) = -2(3)(-1) = 6. The y-intercept is y=6y = 6. Choice A incorrectly identifies the zeros as x=3,−1x = 3, -1 by confusing the signs in the factored form. When you see (x+3)(x + 3), the zero is x=−3x = -3, not x=3x = 3. Choice B makes the same sign error with the zeros and also gets the wrong y-intercept. Choice C correctly identifies the zeros but calculates the y-intercept as y=−6y = -6, likely by making an arithmetic error when multiplying −2(3)(−1)-2(3)(-1). Choice D correctly identifies both the zeros at x=−3,1x = -3, 1 and the y-intercept at y=6y = 6. Remember: in factored form (x−h)(x - h), the zero is x=hx = h. Watch the signs carefully, and always substitute x=0x = 0 to find the y-intercept.

Question 18

The expression 2x2+5x−32x^2 + 5x - 3 represents the net change in a company's stock value. When can this expression be factored, and what do the zeros represent in this context?

  1. Factors as 2(x+3)(x−12)2(x + 3)(x - \frac{1}{2}); zeros indicate when stock value change is neutral
  2. Factors as (2x−1)(x+3)(2x - 1)(x + 3); zeros indicate when stock value change is neutral (correct answer)
  3. Cannot be factored with integer coefficients; no meaningful interpretation of zeros
  4. Factors as (2x+1)(x−3)(2x + 1)(x - 3); zeros indicate maximum and minimum stock values
Explanation: To factor 2x2+5x−32x^2 + 5x - 3, look for factors of 2⋅(−3)=−62 \cdot (-3) = -6 that add to 5: these are 6 and -1. Rewrite: 2x2+6x−x−3=2x(x+3)−1(x+3)=(2x−1)(x+3)2x^2 + 6x - x - 3 = 2x(x + 3) - 1(x + 3) = (2x - 1)(x + 3). Setting equal to zero: 2x−1=02x - 1 = 0 gives x=12x = \frac{1}{2}, and x+3=0x + 3 = 0 gives x=−3x = -3. The zeros represent when the net change is zero (neutral). Choice A has incorrect factorization. Choice C is wrong since it can be factored. Choice D has wrong factors and misinterprets what zeros represent.

Question 19

The quadratic g(x)=x2−6x+8g(x) = x^2 - 6x + 8 models the height of a ball above a reference point. Based on its factored form, when is the ball at the reference level, and what is the axis of symmetry?

  1. At reference level when x=1,8x = 1, 8; axis of symmetry at x=4.5x = 4.5
  2. At reference level when x=−2,−4x = -2, -4; axis of symmetry at x=−3x = -3
  3. At reference level when x=2,4x = 2, 4; axis of symmetry at x=6x = 6
  4. At reference level when x=2,4x = 2, 4; axis of symmetry at x=3x = 3 (correct answer)
Explanation: When you encounter a quadratic function modeling real-world scenarios like projectile motion, you need to find where the function equals zero (the reference level) and identify the axis of symmetry. The key is working with the factored form of the quadratic. To find when the ball is at reference level, you need to factor g(x)=x2−6x+8g(x) = x^2 - 6x + 8. Look for two numbers that multiply to 8 and add to -6. Those numbers are -2 and -4, so the factored form is g(x)=(x−2)(x−4)g(x) = (x - 2)(x - 4). Setting this equal to zero: (x−2)(x−4)=0(x - 2)(x - 4) = 0, which gives x=2x = 2 and x=4x = 4. The axis of symmetry for any parabola is the vertical line halfway between the x-intercepts. Using the formula x=x1+x22x = \frac{x_1 + x_2}{2}, you get x=2+42=3x = \frac{2 + 4}{2} = 3. Answer A incorrectly factors the quadratic, finding roots that don't satisfy the original equation. Answer B uses the factors (-2, -4) as the roots instead of recognizing that if (x−a)=0(x - a) = 0, then x=ax = a. This is a common sign error. Answer C correctly identifies the x-intercepts but miscalculates the axis of symmetry, possibly confusing it with the coefficient of the x-term. Remember: when factoring x2+bx+cx^2 + bx + c, find the x-intercepts by setting each factor equal to zero, and always verify your axis of symmetry is exactly halfway between the roots.

Question 20

The quadratic function f(x)=2x2−8x+6f(x) = 2x^2 - 8x + 6 can be written in factored form. What are the x-coordinates of the vertex and the x-intercepts of this function?

  1. Vertex at x=2x = 2; x-intercepts at x=1x = 1 and x=3x = 3 (correct answer)
  2. Vertex at x=4x = 4; x-intercepts at x=1x = 1 and x=6x = 6
  3. Vertex at x=2x = 2; x-intercepts at x=−1x = -1 and x=−3x = -3
  4. Vertex at x=1x = 1; x-intercepts at x=1x = 1 and x=3x = 3
Explanation: First factor: 2x2−8x+6=2(x2−4x+3)=2(x−1)(x−3)2x^2 - 8x + 6 = 2(x^2 - 4x + 3) = 2(x - 1)(x - 3). The x-intercepts occur where f(x)=0f(x) = 0, so x=1x = 1 and x=3x = 3. The vertex x-coordinate is the midpoint: x=1+32=2x = \frac{1 + 3}{2} = 2. Choice B incorrectly uses x=−ba=82=4x = -\frac{b}{a} = \frac{8}{2} = 4 without factoring out the 2 first. Choice C uses the wrong signs for the intercepts. Choice D confuses the vertex location with an intercept.