An amusement park ride takes riders to the top of a tower ...

9. CCSS MODELING An amusement park ride takes riders to the top of a tower and drops them at speeds reaching

80 feet per second. A function that models this ride is h = –16t2 - 64t – 60, where h is the height in feet and t is the

time in seconds. About how many seconds does it take for riders to drop from 60 feet to 0 feet?

SOLUTION: Substitute 0 for h in the given function.

Identify a, b, and c from the equation.

Substitute these values into the Quadratic Formula and simplify.

t = 0.78 is reasonable.

Solve each equation by using the Quadratic Formula.

SOLUTION:

Write the equation in the form ax2 + bx + c = 0 and identify a, b, and c.

SOLUTION:

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.

SOLUTION: a. Identify a, b, and c from the equation. a = 2, b = 3 and c = –3.

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.

SOLUTION: a. Identify a, b, and c from the equation. a = 4, b = –6 and c = 2.

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.

SOLUTION: a. Identify a, b, and c from the equation. a = 6, b = –1 and c = –5.

Substitute the values in and simplify

SOLUTION: a. Identify a, b, and c from the equation. a = 2, b = 4 and c = 7.

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.

SOLUTION:

a. Write the equation in the form ax2 + bx + c = 0 and identify a, b, and c.

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.

SOLUTION:

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.

SOLUTION:

41. SMOKING A decrease in smoking in the United States has resulted in lower death rates caused by lung cancer.

The number of deaths per 100,000 people y can be approximated by y = –0.26x2 – 0.55x + 91.81, where x

represents the number of years after 2000. a. Calculate the number of deaths per 100,000 people for 2015 and 2017. b. Use the Quadratic Formula to solve for x when y = 50. c. According to the quadratic function, when will the death rate be 0 per 100,000? Do you think that this prediction is reasonable? Why or why not?

SOLUTION:

a. Substitute 15 for x in the equation y = –0.26x2 – 0.55x + 91.81 and simplify.

Substitute 17 for x in the equation y = –0.26x2 – 0.55x + 91.81 and simplify.

b. Substitute 50 for y in the equation and simplify.

a = –0.26, b = –0.55 and c = 41.81. Substitute the values of a, b, and c into the Quadratic Formula and simplify.

The number of deaths cannot have negative value so x is 11.67. c. Substitute 0 for y in the equation.

The year at which the death will be 0 is 2018. No, this prediction is not reasonable. The death rate from cancer will never be 0 unless a cure is found. If and when a cure will be found cannot be predicted.

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?

SOLUTION:

Substitute 666 for S in the formula

a = 1, b = 1, c = –1332. Substitute the values of a, b, and c into the Quadratic Formula and simplify.

So there are 36 consecutive integers.

43. CCSS CRITIQUE Tama and Jonathan are determining the number of solutions of 3x2 – 5x = 7. Is either of them

correct? Explain your reasoning.

SOLUTION:

Jonathan is correct; you must first write the equation in the form ax2 + bx + c = 0 to determine the values of a, b,

and c. Therefore, the value of c is –7, not 7.

44. CHALLENGE Find the solutions of 4ix2 – 4ix + 5i = 0 by using the Quadratic Formula.

SOLUTION: 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.

45. REASONING Determine whether each statement is sometimes, always, or never true. Explain your reasoning. a. In a quadratic equation in standard form, if a and c are different signs, then the solutions will be real. b. If the discriminant of a quadratic equation is greater than 1, the two roots are real irrational numbers.

SOLUTION: a. Always; when a and c are opposite signs, then ac will always be negative and –4ac will always be positive. Since

b2 will also always be positive, then b

2 – 4ac represents the addition of two positive values, which will never be

negative. Hence, the discriminant can never be negative and the solutions can never be imaginary.

b. Sometimes; the roots will only be irrational if b2 – 4ac is not a perfect square.

47. CHALLENGE Find the value(s) of m in the quadratic equation x2 + x + m + 1 = 0 such that it has one solution.

SOLUTION: Since the equation has one solution, its discriminant value is 0.

49. A company determined that its monthly profit P is given by P = –8x2 + 165x – 100, where x is the selling price for

each unit of product. Which of the following is the best estimate of the maximum price per unit that the company cancharge without losing money? A $10 B $20 C $30 D $40

SOLUTION:

Substitute 0 for P in the equation and solve it using Quadratic Formula.

Thus, the best estimate of the maximum price per unit is $20. B is the correct option.

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}

SOLUTION: Only the set {4, 5, 6, 7, 9} has the mean greater than the median. So, H is the correct option.

51. SHORT RESPONSE In the figure below, P is the center of the circle with radius 15 inches. What is the area

SOLUTION:

Substitute 15 for base and 15 for height in the formula

The area of is 112.5 in2.

52. 75% of 88 is the same as 60% of what number? A 100 B 101 C 108 D 110

SOLUTION: Let the unknown number be x.

So, D is the correct option.

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4-6 The Quadratic Formula and the Discriminant

SOLUTION:

An amusement park ride takes riders to the top of a tower ...

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