4-6 The Quadratic Formula and the Discriminant. Solve each equation by using the Quadratic Formula.
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1 2. Solve each equation by using the Quadratic Formula. Write the equation in the form ax 2 + bx + c = 0 and identify a, b, and c. Substitute these values into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 1
2 4. Identify a, b, and c from the equation. Substitute these values into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 2
3 6. Write the equation in the form ax 2 + bx + c = 0 and identify a, b, and c. Substitute these values into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 3
4 8. Write the equation in the form ax 2 + bx + c = 0 and identify a, b, and c. Substitute these values into the Quadratic Formula and simplify. Complete parts a and b for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. 10. a. Identify a, b, and c from the equation. a = 3, b = 8 and c = 2. Substitute the values in and simplify. b. The discriminant is not a perfect square, so there are two irrational roots. esolutions Manual - Powered by Cognero Page 4
5 12. a. Identify a, b, and c from the equation. a = 16, b = 8 and c = 1. Substitute the values in and simplify. b. The discriminant is 0, so there is one rational root. Solve each equation by using the Quadratic Formula. 14. Write the equation in the form ax 2 + bx + c = 0 and identify a, b, and c. Substitute these values into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 5
6 16. Write the equation in the form ax 2 + bx + c = 0 and identify a, b, and c. Substitute these values into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 6
7 18. Write the equation in the form ax 2 + bx + c = 0 and identify a, b, and c. Substitute these values into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 7
8 20. DIVING Competitors in the 10-meter platform diving competition jump upward and outward before diving into the pool below. The height h of a diver in meters above the pool after t seconds can be approximated by the equation h = 4.9t 2 + 3t a. Determine a domain and range for which this function makes sense. b. When will the diver hit the water? a. D: {t 0 t 2}, R: {h 0 h 10} b. Substitute 0 for h in the equation h = 4.9t 2 + 3t Identify a, b, and c from the equation. a = 4.9, b = 3 and c = 10. Substitute the values of a, b, and c into the Quadratic Formula and simplify. As the time cannot be in negative it is about 1.77 seconds. esolutions Manual - Powered by Cognero Page 8
9 22. Complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. a. Identify a, b, and c from the equation. a = 4, b = 6 and c = 2. Substitute the values in and simplify. b. The discriminant is a perfect square, so there are two rational roots. c. Substitute the values of a, b, and c into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 9
10 24. a. Identify a, b, and c from the equation. a = 6, b = 1 and c = 5. Substitute the values in. and simplify b. The discriminant is a perfect square, so there are two rational roots. c. Substitute the values of a, b, and c into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 10
11 26. a. Identify a, b, and c from the equation. a = 2, b = 4 and c = 7. Substitute the values in and simplify. b. The discriminant is negative, so there are two complex roots. c. Substitute the values of a, b, and c into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 11
12 28. a. Write the equation in the form ax 2 + bx + c = 0 and identify a, b, and c. Substitute the values in and simplify. b. The discriminant is 0, so there is one rational root. c. Substitute the values of a, b, and c into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 12
13 30. a. Write the equation in the form ax 2 + bx + c = 0 and identify a, b, and c. Substitute the values in and simplify. b. The discriminant is not a perfect square, so there are two irrational roots. c. Substitute the values of a, b, and c into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 13
14 32. a. Write the equation in the form ax 2 + bx + c = 0 and identify a, b, and c. Substitute the values in and simplify. b. The discriminant is negative, so there are two complex roots. c. Substitute the values of a, b, and c into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 14
15 34. CCSS SENSE-MAKING Civil engineers are designing a section of road that is going to dip below sea level. The road s curve can be modeled by the equation y = x x, where x is the horizontal distance in feet between the points where the road is at sea level and y is the elevation. The engineers want to put stop signs at the locations where the elevation of the road is equal to sea level. At what horizontal distances will they place the stop signs? Substitute 0 for y in the equation y = x x. Identify a, b, and c from the equation. a = , b = 0.06 and c = 0. Substitute the values of a, b, and c into the Quadratic Formula and simplify. The engineers will place the stop signs at 0 ft and 1200 ft. esolutions Manual - Powered by Cognero Page 15
16 Complete parts a c for each quadratic equation. a. Find the value of the discriminant. b. Describe the number and type of roots. c. Find the exact solutions by using the Quadratic Formula. 36. a. Write the equation in the form ax 2 + bx + c = 0 and identify a, b, and c. Substitute the values in and simplify. b. The discriminant is a perfect square, so there are two rational roots. c. Substitute the values of a, b, and c into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 16
17 38. a. Write the equation in the form ax 2 + bx + c = 0 and identify a, b, and c. Substitute the values in and simplify. b. The discriminant is negative, so there are two complex roots. c. Substitute the values of a, b, and c into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 17
18 40. a. Write the equation in the form ax 2 + bx + c = 0 and identify a, b, and c. Substitute the values in and simplify. b. The discriminant is not a perfect square, so there are two irrational roots. c. Substitute the values of a, b, and c into the Quadratic Formula and simplify. esolutions Manual - Powered by Cognero Page 18
19 42. NUMBER THEORY The sum S of consecutive integers 1, 2, 3,, n is given by the formula How many consecutive integers, starting with 1, must be added to get a sum of 666? Substitute 666 for S in the formula a = 1, b = 1, c = Substitute the values of a, b, and c into the Quadratic Formula and simplify. So there are 36 consecutive integers. esolutions Manual - Powered by Cognero Page 19
20 44. CHALLENGE Find the solutions of 4ix 2 4ix + 5i = 0 by using the Quadratic Formula. Identify a, b, and c from the equation. a = 4i, b = 4i and c = 5i. Substitute the values of a, b, and c into the Quadratic Formula and simplify. 46. OPEN ENDED Sketch the corresponding graph and state the number and type of roots for each of the following. a. b 2 4ac = 0 b. A quadratic function in which f (x) never equals zero. c. A quadratic function in which f (a) = 0 and f (b) = 0;. d. The discriminant is less than zero. e. a and b are both solutions and can be represented as fractions. a. Sample answer: 1 rational root b. Sample answer: 2 complex roots esolutions Manual - Powered by Cognero Page 20
21 c. Sample answer: 2 real roots d. Sample answer: 2 complex roots e. Sample answer: 2 rational roots esolutions Manual - Powered by Cognero Page 21
22 48. WRITING IN MATH Describe three different ways to solve x 2 2x 15 = 0. Which method do you prefer, and why? (1) Factor x 2 2x 15 as (x + 3)(x 5). Then according to the Zero Product Property, either x + 3 = 0 or x 5 = 0. Solving these equations, x = 3 or x = 5. (2) Rewrite the equation as x 2 2x = 15. Then add 1 to each side of the equation to complete the square on the left side. Then (x 1) 2 = 16. Taking the square root of each side, x 1 = ± 4. Therefore, x = 1 ± 4 and x = 3 or x = 5. (3) Use the Quadratic Formula. Thus, or Simplifying the expression, x = 3 or x = 5. See students preferences. 50. SAT/ACT For which of the following sets of numbers is the mean greater than the median? F {4, 5, 6, 7, 8} G {4, 6, 6, 6, 8} H {4, 5, 6, 7, 9} J {3, 5, 6, 7, 8} K {2, 6, 6, 6, 6} Only the set {4, 5, 6, 7, 9} has the mean greater than the median. So, H is the correct option. esolutions Manual - Powered by Cognero Page 22
23 52. 75% of 88 is the same as 60% of what number? A 100 B 101 C 108 D 110 Let the unknown number be x. So, D is the correct option. Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. 54. Find one half of 2.4 and square the result. Add the result 1.44 to x x. The value of the c is The trinomial can be written as (x + 1.2) 2. esolutions Manual - Powered by Cognero Page 23
24 Simplify esolutions Manual - Powered by Cognero Page 24
25 60. BUSINESS Ms. Larson owns three fruit farms on which she grows apples, peaches, and apricots. She sells apples for $22 a case, peaches for $25 a case, and apricots for $18 a case. a. Write an inventory matrix for the number of cases for each type of fruit for each farm and a cost matrix for the price per case for each type of fruit. b. Find the total income of the three fruit farms expressed as a matrix. c. What is the total income from all three fruit farms? a. Inventory matrix: Cost matrix: b. c. Total income from all the three fruit farms: = 14, , ,295 = $31,850 esolutions Manual - Powered by Cognero Page 25
26 Write an equation for each graph. 62. This is a parabola with vertex at (0, 0), through the point (4, 4). The equation of the given graph is y = 0.25x 2. esolutions Manual - Powered by Cognero Page 26
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