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9.5: Factoring Quadratic Expressions

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

At the end of this lesson, students will be able to:

  • Write quadratic equations in standard form.
  • Factor quadratic expressions for different coefficient values.
  • Factor when a=1.

Vocabulary

Terms introduced in this lesson:

quadratic polynomial
quadratic trinomials

Teaching Strategies and Tips

In this lesson, students learn to factor quadratic polynomials according to the signs of a,b, and c:

  • a=1,b>0,c>0. See Examples 1-4.

Additional Examples:

Factor.

a. x2+15x+26. Answer: (x+13)(x+2)

b. x2+13x+40. Answer: (x+8)(x+5)

c. x2+20x+75. Answer: (x+15)(x+5)

  • a=1,b<0,c>0. See Examples 5 and 6.

Additional Examples:

Factor.

a. x217x+42. Answer: (x14)(x3)

b. x221x+90. Answer: (x15)(x6)

c. x214x+48. Answer: (x6)(x8)

  • a=1,c<0. See Examples 7-9.

Additional Examples:

Factor.

a. x215x54. Answer: (x18)(x+3)

b. x2+7x60. Answer: (x+12)(x5)

c. x216x192. Answer: (x24)(x+8)

  • a=1. See Example 10.

Additional Examples:

Factor.

a. x24x+60. Answer: (x6)(x+10)

b. x2+14x40. Answer: (x10)(x4)

c. x225x156. Answer: (x+12)(x+13)

  • Allow students to infer that if c>0(a=1), then the factorization will be either of the form (+)(+) or ()() (same signs). If c<0 (a=1), then use the form ()() (different signs).
  • See summary at the end of the lesson for a list of procedures and examples for each case.

Emphasize that factoring is the reverse of multiplication.

  • Use an example such as (x+3)(x+7)=x2+10x+21 in which the binomials are expanded one step at a time to motivate factoring.
  • Demonstrate that factoring is equivalent to putting squares and rectangles back together into larger rectangles.

Example:

Multiply.

(x+3)(x+7).

Solution. The diagram shows that (x+3)(x+7)=x2+10x+21. Observe that it also shows how to factor x2+10x+21.

Suggest that students stop listing the possible products for c after the correct choice is evident.

Error Troubleshooting

General Tip: For quadratic trinomials with a=1, remind students to factor 1 from every term. Remind students to include it in their final answer.

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CK.MAT.ENG.TE.1.Algebra-I.9.5