Lesson 17: The Zero Product Property Revisitedmrpunpanichgulmath.weebly.com/uploads/3/7/5/3/... ·...

ALGEBRA I

Hart Interactive – Algebra 1 M4 Lesson 17

Lesson 17: The Zero Product Property Revisited

Opening Exercise

1. A physics teacher put a ball at the top of a ramp and let it roll

down toward the floor. The class determined that the height of

the ball could be represented by the equation ℎ = −16𝑡𝑡2 + 4,

where the height, ℎ, is measured in feet from the ground and

time, 𝑡𝑡, is measured in seconds.

A. What do you notice about the structure of the quadratic

expression in this problem?

B. In the equation, explain what the 4 represents.

C. Explain how you would use the equation to determine the time it takes the ball to reach the floor.

D. Now consider the possible solutions for 𝑡𝑡. Which values are reasonable? What values would make

no sense in this situation?


Unit 9: More with Quadratics – Factored Form

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ALGEBRA I


Throughout this unit, we’ve been putting equations into factored form so that we could find the x-intercepts.

2. What is the y-value of all x-intercepts? Are there any exceptions?

The x-intercepts give rise to the Zero Product Property. We’ve used this property since Lesson 11, but now

we’ll explore it more analytically.

3. Consider the equation 𝑎𝑎 ⋅ 𝑏𝑏 ⋅ 𝑐𝑐 ⋅ 𝑑𝑑 = 0. What values of 𝑎𝑎, 𝑏𝑏, 𝑐𝑐, and 𝑑𝑑 would make the equation true?

Find values of 𝒄𝒄 and 𝒅𝒅 that satisfy each of the following equations. (There may be more than one correct answer.)

4. 𝑐𝑐𝑑𝑑 = 0 5. (𝑐𝑐 − 5)𝑑𝑑 = 2

6. (𝒄𝒄 − 𝟓𝟓)𝒅𝒅 = 𝟎𝟎 7. (𝒄𝒄 − 𝟓𝟓)(𝒅𝒅 + 𝟑𝟑) = 𝟎𝟎



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ALGEBRA I


8. The equation in the Opening Exercise, ℎ = −16𝑡𝑡2 + 4, describes the height (in feet) of the ball at

different times (in seconds). If we want to find the time when the ball reaches the ground, we would set

h = 0. There are two ways you could approach the problem 0 = −16t2 + 4 as shown below. Complete

each method’s example.

Factoring Method Square Root Method

0 = −16t2 + 4

0 = (−4t + 2)(4t + 2)

−4t + 2 = 0 OR 4t + 2 = 0

t = _____ OR t = _____

0 = −16t2 + 4

16t2 = 4

216 4

16 16

t=

2 1

4t =

1

4t = ±

t = _____ OR t = _____



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ALGEBRA I


Back in Lessons 10 and 11, you explored the licorice-packaging problem. Recall, Isabella wrote the equation for

the volume of the sleeve in vertex form as

2

524 150

2V h= − − + .

9. Isabella’s work to find the horizontal intercepts is shown below. Explain what Isabella did at each step.

2

524 150

2V h= − − +

2

50 24 150

2h= − − + ______________________________________________

2

5150 24

2h− = − − ______________________________________________

2

56.25

2h= − ______________________________________________

2

56.25

2h± = − ______________________________________________

52.5

2h± = − ______________________________________________

52.5

2h± = ______________________________________________

2.5 2.5 h± = ______________________________________________

2.5 2.5 2.5 2.5

5 0

h OR h

h OR h

+ = − == =

______________________________________________



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ALGEBRA I


Additional Practice – Solving for Horizontal Intercepts

Isabella solved for h to find the horizontal intercepts. Use her method to find the x-intercepts for each of the

exercises below. Remember, y is zero when the function hits the x-axis.

10. y = 3(x – 2)2 – 27 11. y = 5(x + 1)

2 – 20

12. y = (x – 7)2 – 25

13. What is the most difficult part of solving for the x-intercepts? Explain your thinking.

Practice Exercises 14 – 20

Solve. Show your work.

14. (x – 2)2 = 16 15. (2x + 4)

2 = 36



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ALGEBRA I


16. 3x2 + 5x + 2 = 0 17. x

2 – 9x = −18

18. 𝑥𝑥2 − 11𝑥𝑥 + 19 = −5 19. 7𝑥𝑥2 + 𝑥𝑥 = 0

20. 7𝑟𝑟2 − 14𝑟𝑟 = −7 21. 2𝑑𝑑2 + 5𝑑𝑑 = 12

22. x2 + 4 = 29 23. 2x

2 = 72

24. 3(x – 3)2 = 27 25. 5(x – 5)

2 + 2 = 22



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ALGEBRA I


Lesson Summary

When solving for the variable in a quadratic equation, rewrite the quadratic expression in factored form and set equal to zero. Using the zero product property, you know that if one factor is equal to zero, then the product of all factors is equal to zero.

Going one step further, when you have set each binomial factor equal to zero and have solved for the variable, all of the possible solutions for the equation have been found. Given the context, some solutions may not be viable, so be sure to determine if each possible solution is appropriate for the problem.

Solving for the x-intercepts Steps

y = (x + 1)2 – 16

0 = (x + 1)2 – 16

16 = (x + 1)2

216 ( 1)x± = +

±4 = x + 1

4 = x + 1 or -4 = x + 1

x = 3 or x = - 5

26. Write in the steps to solve for the x-intercepts.

Zero Product Property

If 𝒂𝒂𝒂𝒂 = 𝟎𝟎, then 𝒂𝒂 = 𝟎𝟎 or 𝒂𝒂 = 𝟎𝟎 or 𝒂𝒂 = 𝒂𝒂 = 𝟎𝟎.



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ALGEBRA I


Homework Problem Set

Solve the following equations.

1. (2𝑥𝑥 − 1)(𝑥𝑥 + 3) = 0 2. (𝑡𝑡 − 4)(3𝑡𝑡 + 1)(𝑡𝑡 + 2) = 0

3. 𝑥𝑥2 − 9 = 0 4. (𝑥𝑥2 − 9)(𝑥𝑥2 − 100) = 0

5. 𝑥𝑥2 − 9 = (𝑥𝑥 − 3)(𝑥𝑥 − 5) 6. 𝑥𝑥2 + 𝑥𝑥 − 30 = 0

7. 𝑝𝑝2 − 7𝑝𝑝 = 0 8. 𝑝𝑝2 − 7𝑝𝑝 = 8

9. 3𝑥𝑥2 + 6𝑥𝑥 + 3 = 0 10. 2𝑥𝑥2 − 9𝑥𝑥 + 10 = 0



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ALGEBRA I


11. 𝑥𝑥2 + 15𝑥𝑥 + 40 = 4 12. 7𝑥𝑥2 + 2𝑥𝑥 = 0

13. 7𝑥𝑥2 + 2𝑥𝑥 − 5 = 0 14. 𝑏𝑏2 + 5𝑏𝑏 − 35 = 3𝑏𝑏

15. 6𝑟𝑟2 − 12𝑟𝑟 = 18 16. 2𝑥𝑥2 + 11𝑥𝑥 = 𝑥𝑥2 − 𝑥𝑥 − 32

17. Write an equation (in factored form) that has solutions of 𝑥𝑥 = 2 or 𝑥𝑥 = 3.

18. Write an equation (in factored form) that has solutions of 𝑎𝑎 = 0 or 𝑎𝑎 = −1.



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ALGEBRA I


19. Quinn looks at the equation (𝑥𝑥 − 5)(𝑥𝑥 − 6) = 2 and says that since the equation is in factored form it

can be solved as follows:

(𝑥𝑥 − 5)(𝑥𝑥 − 6) = 2

𝑥𝑥 − 5 = 2 or 𝑥𝑥 − 6 = 2

𝑥𝑥 = 7 or 𝑥𝑥 = 8.

Explain to Quinn why this is incorrect. Show her the correct way to solve the equation.

For each problem, determine the x-intercepts, the vertex and the y-intercept.

20. y = –2(x + 5)2 + 18

x- intercepts: _____ and _____

vertex: ( _____, _____)

y-intercept: ______

21. y = – (x – 5)2


vertex: ( _____, _____)

y-intercept: ______

22. y = 1

3(x – 1)

2 – 3


vertex: ( _____, _____)

y-intercept: ______



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ALGEBRA I


23. y = 3

4(x + 4)

2 –

27

16


vertex: ( _____, _____)

y-intercept: ______

24. y = 1

3− (x + 1)

2 + 3


vertex: ( _____, _____)

y-intercept: ______

25. y = 4(x + 2)2 – 4


vertex: ( _____, _____)

y-intercept: ______

26. y = – (x + 2)2 + 25


vertex: ( _____, _____)

y-intercept: ______

27. y = 1

2(x + 3)

2 – 18


vertex: ( _____, _____)

y-intercept: ______

28. y = 1

3− (x – 1)

2 + 27


vertex: ( _____, _____)

y-intercept: ______



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ALGEBRA I




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