lesson 13: more factoring strategies for quadratic...

ALGEBRA I

Hart Interactive – Algebra 1 M4 Lesson 13

Lesson 13: More Factoring Strategies for Quadratic Equations & Expressions

Opening Exploration – Looking for Signs

In the last lesson, we focused on quadratic equations where all the terms were positive. Juan’s examples had one with mixed signs though. Let’s see what happens when the signs are positive, negative or a mixture of both.

Use the Factor Bank to help you factor each exercise below.

1. y = x2 + 5x + 6 2. y = x2 – 5x + 6

3. y = x2 – 5x – 6 4. y = x2 + 5x – 6

Discussion

5. What patterns do you notice in these four exercises? What generalizations can we make?

Factor Bank

6 -6 1 • 6 -1 • 6 (-1) • (-6) 1 • (-6) 2 • 3 -2 • 3 (-2) • (-3) 2 • (-3)

Lesson 13: More Factoring Strategies for Quadratic Equations & Expressions Unit 11: More with Quadratics – Factored Form

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ALGEBRA I


Create a Factor Bank for the next four problems. Then factor each one.

6. y = x2 + 9x + 8 7. y = x2 – 9x + 8

8. y = x2 – 7x – 8 9. y = x2 + 7x – 8

Discussion

10. Did your patterns from Exercise 5 hold for these problems?

11. If the last term (the c value) is positive, what must be true about the two factors you use for the middle term (bx)?

12. If the last term (the c value) is negative, what must be true about the two factors you use for the middle term (bx)?

Factor Bank

8 -8


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ALGEBRA I


Practice

Factor the following quadratic expressions.

13. 2𝑥𝑥2 + 10𝑥𝑥 + 12 14. 6𝑥𝑥2 + 5𝑥𝑥 − 6

15. x2 – 12x + 20 16. x2 – 21x – 22 17. 2𝑥𝑥2 − 𝑥𝑥 − 10 18. 6𝑥𝑥2 + 7𝑥𝑥 − 20 19. x2 – 2x – 15 20. x2 + 2x – 15

21. How can you check that your factorization is correct?


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ALGEBRA I


Exploration – A Special Case 22. Consuela ran across the quadratic equation y = 4x2 – 16 and wondered how it could be factored. She

rewrote it as y = 4x2 + 0x – 16.

A. Use the patterns you’ve discovered in this lesson and the last lesson to factor this quadratic function.

B. What are the key features of the parabola’s graph?

x-intercepts: _____, _____ y-intercept: _____

vertex: (_____, _____)

C. Graph the quadratic in the grid at the right.

Factor the following quadratic functions. Use Consuela’s idea of a 0 middle term if necessary. Look for patterns as you go. 23. y = x2 – 9 24. y = x2 – 25

25. What is the generic rule for factoring a quadratic in the form a2 – b2?

-20-19-18-17-16-15-14-13-12-11-10

-9-8-7-6-5-4-3-2-10123456789

1011121314151617181920

-3 -2 -1 0 1 2 3


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ALGEBRA I


Homework Problem Set

1. Equations of the form a2 – b2 are called the difference of perfect squares. Why do you think they have that name?

Factor the following examples of the difference of perfect squares. Notice that these are not written as quadratic functions. They are simply expressions – we could not graph them or tell their key features.

2. 𝑡𝑡2 − 25 3. 4𝑥𝑥2 − 9

4. 16ℎ2 − 36𝑘𝑘2 5. 4 − 𝑏𝑏2

6. 𝑥𝑥4 − 4 7. 𝑥𝑥6 − 25

Challenge: Factor each of the following differences of squares completely:

8. 9𝑦𝑦2 − 100𝑧𝑧2 9. 𝑎𝑎4 − 𝑏𝑏6

10. Challenge 𝑟𝑟4 − 16𝑠𝑠4 (Hint: This one factors twice.)


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ALGEBRA I


11. For each of the following, factor out the greatest common factor.

a. 6𝑦𝑦2 + 18 b. 27𝑦𝑦2 + 18𝑦𝑦 c. 21𝑏𝑏 − 15𝑎𝑎 d. 14𝑐𝑐2 + 2𝑐𝑐 e. 3𝑥𝑥2 − 27

12. The measure of a side of a square is 𝑥𝑥 units. A new square is formed with each side 6 units longer than the original square’s side. Write an expression to represent the area of the new square. (Hint: Draw the new square and count the squares and rectangles.)

Original Square 𝑥𝑥


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𝑥𝑥 + 3

3𝑥𝑥 − 4

3𝑥𝑥 + 4

𝑥𝑥 − 3

13. In the accompanying diagram, the width of the inner rectangle is represented by 𝑥𝑥 − 3 and the length by 𝑥𝑥 + 3. The width of the outer rectangle is represented by 3𝑥𝑥 − 4 and the length by 3𝑥𝑥 + 4. a. Write an expression to represent the area of the larger rectangle. b. Write an expression to represent the area of the smaller rectangle.

MIXED REVIEW

Factor completely.

14. 9𝑥𝑥2 − 25𝑥𝑥 15. 9𝑥𝑥2 − 25 16. 9𝑥𝑥2 − 30𝑥𝑥 + 25 17. 2𝑥𝑥2 + 7𝑥𝑥 + 6 18. 6𝑥𝑥2 + 7𝑥𝑥 + 2 19. 8𝑥𝑥2 + 20𝑥𝑥 + 8

20. 3𝑥𝑥2 + 10𝑥𝑥 + 7 21. 4𝑥𝑥2 + 4𝑥𝑥 + 1


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22. The area of the rectangle at the right is represented by the expression 18𝑥𝑥2 + 12𝑥𝑥 + 2 square units. Write two expressions to represent the dimensions, if the length is known to be twice the width.

23. Two mathematicians are neighbors. Each owns a separate rectangular plot of land that shares a boundary and has the same dimensions. They agree that each has an area of 2𝑥𝑥2 + 3𝑥𝑥 + 1 square units. One mathematician sells his plot to the other. The other wants to put a fence around the perimeter of his new combined plot of land. How many linear units of fencing does he need? Write your answer as an expression in 𝑥𝑥.

Note: This question has two correct approaches and two different correct solutions. Can you find them both?

𝟏𝟏𝟏𝟏𝒙𝒙𝟐𝟐 + 𝟏𝟏𝟐𝟐𝒙𝒙 + 𝟐𝟐


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lesson 13: more factoring strategies for quadratic...

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