Solving a Quadratic Equation by Factoring. What are the solutions of the quadratic equation x2 â 5x + 6 = 0? (x - 2)(x - 3) = 0. Factor the quadrati...

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Quadratic Equations

TEKS FOCUS

VOCABULARY

ĚZero of a function – A zero of a function f(x) is

TEKS (4)(F) Solve quadratic and square root equations. TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. Additional TEKS (1)(A), (1)(E), (7)(I)

any value for which f(x) = 0.

ĚZero-Product Property – If the product of two or more factors is zero, then one of the factors must be zero.

ĚNumber sense – the understanding of what numbers mean and how they are related

ESSENTIAL UNDERSTANDING To find the zeros of a quadratic function y = ax2 + bx + c, solve the related quadratic equation 0 = ax2 + bx + c.

Property

Zero-Product Property If ab = 0, then a = 0 or b = 0.

Problem 1 P

TEKS Process Standard (1)(E)

Solving a Quadratic Equation by Factoring What do you know about the factors of x2 − bx + c? The product of their constant terms is c. The sum is -b.

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Lesson 5-6

What are the solutions of the quadratic equation x2 − 5x + 6 = 0? W (x - 2)(x - 3) = 0

Factor the quadratic expression.

x-2=0

or

x-3=0

Use the Zero-Product Property.

x=2

or

x=3

The T solutions are x = 2 and x = 3.

Quadratic Equations

Solve for x.

Problem 2 P

TEKS Process Standard (1)(C)

Solving a Quadratic Equation With Tables What are the solutions of the quadratic equation 5x2 + 30x + 14 = 2 − 2x? 5x2 + 30x + 14 = 2 - 2x 5x2 + 32x + 12 = 0

Rewrite in standard form.

Use your calculator’s TABLE feature to find the zeros.

What should you look for in the calculator table? Look for x-values for which y = 0.

Plot1 Plot2 Plot3 \Y1 = 55X X 2+32X+12 \Y2 = \Y3 = \Y4 = Enter the \Y5 = equation in \Y6 = \Y7 = standard form as Y1.

X –6 –5 –4 –3 –2 –1 0

X = -6

Y1 0 –23 23 –36 –39 –32 –15 12

Y1 = 0, x = –6 is one zero.

x-interval changed to .1. X –.8 –.7 –.6 –.5 –.4 –.3 –.2

X = -.4

Y1 –10.4 –7.95 7.95 –5.4 –2.75 –2.7 75 0 2.85 85 5.8

Y1 = 0, x = –.4 is the second zero.

Second zero is between x = –1 and x = 0. Notice change in sign for y-values. The solutions are x = -6 and x = -0.4.

Problem bl 3 Solving a Quadratic Equation by Graphing What are the solutions of the quadratic equation 2x2 + 7x = 15? 2x2 + 7x = 15 2x2 + 7x - 15 = 0

How can you use a graph to find the solutions? Find the zeros of the related quadratic function.

Plot1 Plot2 Plot3 \Y1 = 2X 2+7X–15 \Y2 = \Y3 = Enter the \Y4 = \Y5 = equation in \Y6 = standard form \Y7 = as Y1.

Rewrite in standard form. Use ZERO option in CALC feature.

Zero X=–55

Y=0

Zer Zero eroo X=1.5

Y=0

The solutions are x = -5 and x = 1.5.

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Problem 4 P Using a Quadratic Equation Competition From the time Mark Twain wrote The Celebrated Jumping Frog of Calaveras County in 1865, frog-jumping competitions have been growing in popularity. The graph shows a function modeling the height of one frog’s jump, where x is the distance, in feet, from the jump’s start. A How far did the frog jump?

How can you use a graphing calculator to determine the distance? Graph the function and locate the point where the graph crosses the x-axis.

The height of the jump is 0 at the start and end of the jump. Find the zeros of the function. Use a graphing calculator to find the zeros of the related function y = -0.029x2 + 0.59x. In the air Plot1 Plot2 \Y1 = –.029X 2+.59X \Y2 = \Y3 = \Y4 = \Y5 = \Y6 =

Plot3

Take-off

Land

Nothing past this point has real-world meaning.

x = 0 at take-off

Zero X=20.34

x = 20.34 at landing

Y=0

The frog jumped about 20.34 ft. B How high did the frog jump?

The maximum height of the jump is the maximum value of the function. This occurs midway, at 10.17 ft from the start. Find y for x = 10.17. y = -0.029(10.17)2 + 0.59(10.17) ≈ 3.0 The frog jumped to a height of about 3.0 ft. C What is a reasonable domain and range for such a frog-jumping function?

While the function y = -0.029x2 + 0.59x has a domain of all real numbers, actual frog jumping does not allow negative values. So, a reasonable domain for frog-jumping distances is 0 … x … 30. A reasonable range is 0 … y … 5.

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Lesson 5-6

Quadratic Equations

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PRACTICE and APPLICATION EXERCISES

Scan page for a Virtual Nerd™ tutorial video.

Solve each equation by factoring. Check your answers. For additional support when completing your homework, go to PearsonTEXAS.com.

1. x2 + 6x + 8 = 0

2. x2 + 18 = 9x

3. 2x2 - x = 3

4. x2 - 10x + 25 = 0

5. 6x2 + 4x = 0

6. 2x2 = 8x

Solve each equation using tables. Give each answer to at most two decimal places. 7. x2 + 5x + 3 = 0

8. x2 - 7x = 11

9. 2x2 - x = 2

Solve each equation by graphing. Give each answer to at most two decimal places. 10. 6x2 = -19x - 15

11. 3x2 - 5x - 4 = 0

12. 5x2 - 7x - 3 = 8

13. Evaluate Reasonableness (1)(B) A classmate solves the quadratic equation as shown. Find and correct the error. What are the correct solutions? 14. Create Representations to Communicate Mathematical Ideas (1)(E) Write an equation with the given solutions. a. 3 and 5

b. -3 and 2

x2 + 5x + 6 = 2 (x + 2) (x + 3) = 2 x = -2 or x = -3

c. -1 and -6

15. Use Representations to Communicate Mathematical Ideas (1)(E) The function h = -16t 2 + 1700 gives an object’s height h, in feet, at t seconds. a. What does the constant 1700 tell you about the height of the object? b. What does the coefficient of t 2 tell you about the direction the object is moving? c. When will the object be 1000 ft above the ground? d. When will the object be 940 ft above the ground? e. What are a reasonable domain and range for the function h? 16. Apply Mathematics (1)(A) Suppose you want to put a frame around the painting shown at the right. The frame will be the same width around the entire painting. You have 276 in.2 of framing material. How wide should the frame be? 17. The period of a pendulum is the time the pendulum takes to swing back and forth. The function L = 0.81t 2 relates the length L in feet of a pendulum to the time t in seconds that it takes to swing back and forth. A convention center has a pendulum that is 90 feet long. Find the period.

16 in.

24 in.

18. Apply Mathematics (1)(A) Suppose you have an outdoor pool measuring 25 ft by 10 ft. You want to add a cement walkway around the pool. If the walkway will be 1 ft thick and you have 304 ft3 of cement, how wide should the walkway be?

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Select Tools to Solve Problems (1)(C) Solve each equation by factoring, using tables, or by graphing. If necessary, round your answer to the nearest hundredth. 19. x2 + 2x = 6 - 6x

20. 6x2 + 13x + 6 = 0

21. 2x2 + x - 28 = 0

22. (x + 3)2 = 9

23. x2 + 4x = 0

24. x2 = 8x - 7

25. x2 - 3x = 6

26. 4x2 + 5x = 4

27. 7x - 3x2 = -10

Connect Mathematical Ideas (1)(F) The graphs of each pair of functions intersect. Find their points of intersection without using a calculator. (Hint: Solve as a system using substitution.) 28. y = x2

29. y = x2 - 2 3

y = - 12x2 + 2x + 3

30. y = -x2 + x + 4

y = 3x2 - 4x - 2

y = 2x2 - 6

31. The equation x2 - 10x + 24 = 0 can be written in factored form as (x - 4)(x - 6) = 0. How can you use this fact to find the vertex of the graph of y = x2 - 10x + 24? 32. a. Let a 7 0. Use algebraic or arithmetic ideas to explain why the lowest point on the graph of y = a(x - h)2 + k must occur when x = h. b. Suppose that the function in part (a) is y = a(x - h)3 + k. Is your reasoning still valid? Explain. 33. Apply Mathematics (1)(A) When serving in tennis, a player tosses the tennis ball vertically in the air. The height h of the ball after t seconds is given by the quadratic function h(t) = -5t 2 + 7t (the height is measured in meters from the point of the toss). a. How high in the air does the ball go? b. Assume that the player hits the ball on its way down when it’s 0.6 m above the point of the toss. For how many seconds is the ball in the air between the toss and the serve?

TEXAS Test Practice T 34. What are the solutions of the equation 6x 2 + 9x - 15 = 0? A. 1, -15

5

F. ( -1, 7)

G. (3, 7)

5

B. 1, - 2 C. -1, -5 D. 3, 2 35. The vertex of a parabola is (3, 2). A second point on the parabola is (1, 7). Which point is also on the parabola? H. (5, 7)

J. (3, -2)

36. For which quadratic function is -3 the constant term? A. y = (3x + 1)( -x - 3)

C. f (x) = (x - 3)(x - 3)

B. y = x 2 - 3x + 3

D. g(x) = -3x 2 + 3x + 9

37. What transformations are needed to go from the parent function f (x) = x2 to the new function g(x) = -3x2 + 2? Graph g(x).

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Lesson 5-6

Quadratic Equations

Activity Lab USE WITH LESSON 5-6

Writing Equations From Roots

TEKS (4)(B), (1)(F)

A root of an equation is a value that makes the equation true. You can use the Zero-Product Property to write a quadratic function from its zeros or a quadratic equation from its roots.

1 1. 1 a. b. 2. a. b. c.

Write W i a nonzero linear function f (x) that has a zero at x = 3. Write a nonzero linear function g(x) that has a zero at x = 4. For f and g from Exercise 1, write the product function h(x) = f (x) What kind of function is h(x)? Solve the equation h(x) = 0.

# g(x).

Mental Math Write a quadratic equation with each pair of values as roots. 3. 5 and 3

4. 2.5 and 4

6. 5 and 10

3 7. 2 and -2

5. -4 and 4

You can also use zeros or roots to write quadratic expressions in standard form.

2

a

8. and complete the table. Write the product 8 a. Copy C (x - a)(x - b) in standard form for each pair a and b. b. Is there a pattern in the table? Explain. 9. a. If you know the roots, you can write a quadratic function or equation in standard form. Explain how. b. Demonstrate your method for each pair of values in Exercises 3–7.

b

aⴙb

ab

(x ⴚ a) (x ⴚ b) x2 ⫺ 9x ⫹ 20

4

5

9

20

⫺4

5

1

⫺20

■

4

⫺5

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⫺4

⫺5

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⫺9

⫺1

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⫺2

7

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Exercises 10. Explain how to write a quadratic equation that has -6 as its only root. 11. Describe the family of quadratic functions that have zeros at r and s. Sketch several members of the family in the coordinate plane. Find the sum and product of the roots for each quadratic equation. 12. 2x 2 + 3x - 2 = 0

13. x 2 - 2x + 1 = 0

14. x 2 - 5x + 6 = 0

Given the sum and product of the roots, write a quadratic equation in standard form. 15. sum = -3, product = -18

16. sum = 4, product = 3

3

17. sum = 2, product = 4

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