8-7 Solving ax^2 + bx + c = 0

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1 29. BASKETBALL When Jerald shoots a free throw, the ball is 6 feet from the floor and has an initial upward velocity of 20 feet per second. The hoop is 10 feet from the floor. a. Use the vertical motion model to determine an equation that models Jerald s free throw. b. How long is the basketball in the air before it reaches the hoop? c. Raymond shoots a free throw that is 5 foot 9 inches from the floor with the same initial upward velocity. Will the ball be in the air more or less time? Explain. a. A model for the vertical motion of a projected object is given by h = 16t 2 + vt + h 0, where h is the height in feet, t is the time in seconds, v is the initial velocity in feet per second, and h 0 is the initial height in feet. The initial velocity of the basketball, v, is 20 feet per second. The initial height of the basketball, h 0, is 6 feet. The height of the hoop, h, is 10 feet. So, the equation 10 = 16t t + 6 models Jerald s free throw. b. c. The ball will be in the air less time because it starts closer to the ground so the shot will not have as far to fall. 30. DIVING Ben dives from a 36-foot platform. The equation h = 16t t + 36 models the dive. How long will it take Ben to reach the water? When Ben reaches the pool, his height, h will be 0. The roots are and 2. Because time cannot be negative, Ben will reach the water after 2 seconds. 31. NUMBER THEORY Six times the square of a number x plus 11 times the number equals 2. What are possible values of x? Let x = a number. Then, 6x 2 +11x = 2. The roots are and 1. The basketball takes second to reach a height of 10 feet on its way up. The basketball takes 1 second to reach a height of 10 feet on its way down. So, the basketball will be in the air 1 second before it reaches the hoop. The possible values of x are 2 or. esolutions Manual - Powered by Cognero Page 1

2 Factor each polynomial, if possible. If the polynomial cannot be factored using integers, write prime x 2 23x x 2 15x 14 4x 2 15x 14 = 1(4x x + 14) Then factor the trinomial 4x x Then factor the trinomial 6x x In this trinomial, a = 6, b = 23 and c = 20, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the positive factors of 6(20) or 120 and identify the factors with a sum of 23. Factors of 120 1, , , , , , , , In this trinomial, a = 4, b = 15 and c = 14, so m + p is positive and mp is positive. Therefore, m and p must both be positive. List the positive factors of 4(14) or 56 and identify the factors with a sum of 15. Factors of 56 1, , , , The correct factors are 7 and 8. The correct factors are 8 and 15. So, 4x 2 15x 14 = (x + 2)(4x + 7). So, 6x 2 23x 20= (2x + 5)(3x + 4). esolutions Manual - Powered by Cognero Page 2

3 34. 5x x + 8 5x x + 8 = 1(5x 2 18x 8) Then factor the trinomial 5x 2 18x 8. In this trinomial, a = 5, b = 18 and c = 8, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 5( 8) or 40 and identify the factors with a sum of 18. Factors of 40 1, , , , , , , 8 3 5, 8 3 The correct factors are 20 and x x 35 6x x 35 = 1(6x 2 31x + 35) Then factor the trinomial 6x 2 31x In this trinomial, a = 6, b = 31 and c = 35, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 6 (35) or 210 and identify the factors with a sum of 31. Factors of 210 1, , , , , , , , The correct factors are 21 and 10. So, 5x x + 8 = (x 4)(5x + 2). So, 6x x 35 = (2x 7)(3x 5). esolutions Manual - Powered by Cognero Page 3

4 36. 4x 2 + 5x 12 4x 2 + 5x 12 = 1(4x 2 5x + 12) Then factor the trinomial 4x 2 5x In this trinomial, a = 4, b = 5 and c = 12, so m + p is negative and mp is positive. Therefore, m and p must both be negative. List the negative factors of 4(12) or 48 and identify the factors with a sum of 5. Factors of 48 1, , , , , 8 14 There are no factors of 48 with a sum of 5. So, 4x 2 + 5x 12 is prime x 2 + x x 2 + x + 20 = 1(12x 2 x 20) Then factor the trinomial 12x 2 x 20. In this trinomial, a = 12, b = 1 and c = 20, so m + p is negative and mp is negative. Therefore, m and p must have different signs. List the factors of 12( 20) or 240 and identify the factors with a sum of 1. Factors of 240 1, , , , , , , , , , , , , , , , , , , , 16 1 The correct factors are 16 and 15. So, 12x 2 + x + 20 = (4x + 5)(3x 4). esolutions Manual - Powered by Cognero Page 4

5 38. URBAN PLANNING The city has commissioned the building of a rectangular park. The area of the park can be expressed as 660x x Factor this expression to find binomials with integer coefficients that represent possible dimensions of the park. If x = 8, what is a possible perimeter of the park? a. The area of a square is found using the formula A = s 2. So, the area of the larger square is a 2 and the area of the smaller square is b 2. b. So, (22x + 5)(30x + 17) represent the possible dimensions of the park. Evaluate each dimension when x = 8 to find the possible length and width of the park when x = 8. To find the area of the remaining region, subtract the area of the smaller square from the area of the larger square. So, the area of the remaining region is a 2 b 2. c. A possible perimeter of the park is 876 units. 39. MULTIPLE REPRESENTATIONS In this problem, you will explore factoring a special type of a. GEOMETRIC Draw a square and label the sides a. Within this square, draw a smaller square that shares a vertex with the first square. Label the sides b. What are the areas of the two squares? b. GEOMETRIC Cut and remove the small square. What is the area of the remaining region? c. ANALYTICAL Draw a diagonal line between the inside corner and outside corner of the figure, and cut along this line to make two congruent pieces. Then rearrange the two pieces to form a rectangle. What are the dimensions? d. ANALYTICAL Write the area of the rectangle as the product of two binomials. e. VERBAL Complete this statement: a 2 b 2 = Why is this statement true? The width is a b and the length is a + b. d. e. The figure with area a 2 b 2 and the rectangle with area (a b)(a + b) have the same area, so a 2 b 2 = (a b)(a + b). esolutions Manual - Powered by Cognero Page 5