Chapter 16 Review. 1. What is the solution set of n 2 + 5n 14 = 0? (A) n = {0, 14} (B) n = { 1, 14} (C) n = { 2, 7} (D) n = { 2, 7} (E) n = { 7, 2}

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1 Chapter 16 Review Directions: For each of the questions below, choose the best answer from the five choices given. 1. What is the solution set of n + 5n 14 = 0? (A) n = {0, 14} (B) n = { 1, 14} (C) n = {, 7} (D) n = {, 7} (E) n = { 7, }. Which of the following choices is equivalent to the equation x 8x 48 = 0? (A) (x + 8) = 64 (B) (x 8) = 16 (C) (x 4) = 16 (D) (x + 4) = 64 (E) (x 4) = The length of Sarita s rectangular garden is 6 feet less than the width. The area of her garden is 16 square feet. Which of the following equations could be used to find the width of Sarita s garden in feet? (A) 4w 1 = 16 (B) w 6w = 0 (C) w + 6w = 16 (D) w 6w 16 = 0 (E) w + w 6 = For which of the following equations are x = and x = 6 both solutions? (A) x + x 1 = 0 (B) x 4x 1 = 0 (C) x 8x + 49 = 0 (D) x + 8x 4 = 0 (E) x + x 4 = 0

2 5. What is the product of the roots of the equation x 9x + 4 = 0? (A) 4 (B) 4 (C) (D) 8 (E) 8 6. If x + 9x =, then one of the possible values of x + 4x is (A) (B) (C) (D) (E) 1 7. On what interval will 0 x be negative? (A) < x < (B) x < or x > (C) x = and x = 0 (D) 0 < x < 0 (E) 0 x will not be negative on any interval. 8. Which of the following choices is a root of x + 7x 5 = 0? (A) (B) (C) (D) (E) 5 9. For which of the following equations is 4 a solution? (A) x x 8 = 0 (B) x 8x 14 = 0 (C) x + 4x + 6 = 0 (D) x 14 = 0 (E) x + 8x + 14 = 0

3 . If 5x 5x + 1 = 0, then one of the possible values of x is (A) 5 (B) 5 (C) (D) 5 (E) The sides of a square are all increased by 5 cm. The area of the new square is 50 cm. How many centimeters is the length of a side of the original square? (A) 0 (B) 5 5 (C) 5+ 5 (D) 5 (E) Which of the following equations has a vertex at the point (8, 0)? (A) y = x 64 (B) y = x 16x 64 (C) y = x 16x + 64 (D) y = x + 64 (E) y = 64 x

4 ANSWERS EXPLAINED 1. (E) To find the solution set, start by factoring the quadratic expression. n + 5n 14 = 0 (n + 7)(n ) = 0 Then, write each binomial as a separate equation equal to 0, and solve. (n + 7) = 0 (n ) = 0 n = 7 n = n = { 7, }. (E) To find the equivalent equation, we may need to expand each answer choice to determine what each one looks like as a quadratic equation. However, looking at x 8x 48 = 0, we know that we want b to equal 8. Looking at the answer choices, only choices C and E will result in a b term of 8x. Therefore, we will expand only choices C and E. Expand and investigate choice C. (x 4) = 16 (x 4)(x 4) = 16 x 8x + 16 = 16 x 8x = 0 Expand and investigate choice E. (x 4) = 64 (x 4)(x 4) = 64 x 8x + 16 = 64 x 8x 48 = 0 The only choice that produces an equivalent expression after expanding it and putting it into standard form is choice E. 3. ( D) Using l to represent the length of Sarita s garden, and w to represent the width of her garden, we could write the equation l w = 16 to represent the area of her garden in square feet. We also know that the length of her garden is 6 feet less than the width, so we can state that l = w 6. We can then rewrite the equation as (w 6) w = 16. This becomes w (w 6) = 16, and then w 6w = 16. To solve a quadratic equation, we normally put the equation in standard form, ax + bx + c = 0. In standard form, w 6w = 16 becomes w 6w 16 = 0, which is choice D. 4. (D) We could approach this problem by separately solving each equation found in the answer choices and finding the one that has x = { 6, } for its solution set. Alternatively, we can separately substitute 6 for x and for x into each equation to determine the equation that equals 0 for both x = 6 and x =.

5 Let s start by substituting for x into each of the answer choices. (A) x + x 1 = + 1 = 6 (B) x 4x 1 = () 4() 1 = 1 (C) x 8x + 49 = () 8() + 49 = 41 (D) x + 8x 4 = () + 8() 4 = 0 (E) x + x 4 = ( ) + () 4 = 1 Only choice D has an equation that equals 0 for x =, so this should be the answer. Let s substitute 6 for x just to be sure. (D) x + 8x 4 = ( 6) + 8( 6) 4 = 0 The correct answer is choice D. 5. (C) First we must find the roots of this equation. We can factor the quadratic expression because ac = 8 and b = 9, and 1 8 = 9, and = 9. Rewrite the quadratic expression by separating the b term into x and 8x. x 9x + 4 = 0 x x 8x + 4 = 0 Then, factor and solve. x(x 1) 4(x 1) = 0 (x 1)(x 4) = 0 x 1 = 0 x 4 = 0 x = 1 x = 4 The product of the roots is found using 1 4 =. 6. (B) Start by putting the complete quadratic equation in standard form. x + 9x = x + 9x + = 0 We can factor this equation because ac = 0 and b = 9, and 4 5 = 0 and = 9. Rewrite the quadratic expression by separating the b term into 4x and 5x. x + 4x + 5x + = 0 Then, factor and solve. x(x + ) + 5(x + ) = 0 (x + 5)(x + ) = 0 x + 5 = 0 x + = 0 x = 5 x =

6 Then, evaluate x + 4x for both x = and x =. ( ) + 4( ) Only choice B contains one of our possible solutions, so that is the correct answer. 7. (B) First, find the real zeros of 0 x by setting the expression equal to zero. Then, solve by factoring. 0 = 0 x 0 = ( x)( + x) Solving each binomial for x, we find that the equation has zeros at x = and x =. We also know that the graph is a parabola that opens downward. It has a vertex at x = 0, which is the midpoint of the zeros. This vertex is the point where the graph of y = 0 x reaches its maximum value. The value of 0 x will be negative at all values of x that lie to the left and right of the zeros. That is, 0 x is negative when x < and when x >. 8. (A) We cannot easily factor this quadratic equation, so identify a, b, and c, and substitute the values into the quadratic equation. a = 1, b = 7, c = (1)( 5) x = ± (1) x = ± x = ± Choice A gives us one of the roots we found. 9. (E) We can either proceed by separately solving each quadratic equation presented as an answer choice, or we can alternatively substitute the given solution into each equation to find the one that gives us 0 for x = 4. Choice A: x x 8 = = = = 0

7 Choice B: Choice C: Choice D: Choice E: x 8x 14 = = = = 0 x + 4x + 6 = = = = 0 ( ) x 14 = = = = 0 x + 8x + 14 = = = 0 0 = 0 Only choice E produces an end result of 0.. (E) First, to determine what x equals, identify a, b, and c from 5x 5x + 1 = 0, and substitute the values into the quadratic formula. 5 ( 5) 4(5)(1) x = ± (5) 5± 5 0 x = 5± 5 x = Then, to determine what x is equal to, we need to evaluate the expression for both possible values of x. x = x = ()() x = 0 x = ( )( ) x = 5 5 x = 5 5 ()() x 30 5 = 0 x = 3 5 ( )( 5 5 )

8 Choice E offers the second value that we found for x, so this is the correct answer choice. 11. (C) Using this information, we can state that if s equals the length of a side of the original square in centimeters, then s + 5 equals the length of a side of the new square. We can then use the area formula for a square to represent the area of the new square as: (s + 5) = 50 cm s + s + 5 = 50 s + s 5 = 0 Then, using the quadratic formula, we can solve for s. 4(1)( 5) x = ± (1) 0 0 x = ± + 00 x = ± x = ± x = 5± 5 Choices B and C each present us with one of the solutions to the quadratic equation s + s 5 = 0. However, 5 5 will result in a negative outcome. We must reject this choice because the length of a side of a square cannot be a negative number of centimeters long. Therefore, choice C is the only correct answer. 1. (C) If the vertex is at (8, 0), then this is also the only real zero of the graph. Only one of the equations, y = x 16x + 64, has just one real zero. Set this equation equal to zero and factor: y = x 16x = (x 8)(x 8) Solving for x, we find that both binomials produce a solution of x = 8.