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9.5: Factoring Quadratic Expressions

Created by: CK-12

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. x^2 + 15x + 26. Answer: (x + 13)(x + 2)

b. x^2 + 13x + 40. Answer: (x + 8)(x + 5)

c. x^2 + 20x + 75. Answer: (x + 15)(x + 5)

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

Additional Examples:

Factor.

a. x^2 - 17x + 42. Answer: (x - 14)(x - 3)

b. x^2 - 21x + 90. Answer: (x - 15)(x - 6)

c. x^2 - 14x + 48. Answer: (x - 6)(x - 8)

  • a = 1, c < 0. See Examples 7-9.

Additional Examples:

Factor.

a. x^2 - 15x - 54. Answer: (x - 18)(x + 3)

b. x^2 + 7x - 60. Answer: (x + 12)(x - 5)

c. x^2 - 16x - 192. Answer: (x - 24)(x + 8)

  • a = -1. See Example 10.

Additional Examples:

Factor.

a. -x^2 - 4x + 60. Answer: -(x - 6)(x + 10)

b. -x^2 + 14x - 40. Answer: -(x - 10)(x - 4)

c. -x^2 - 25x - 156. Answer: -(x + 12)(x + 13)

  • Allow students to infer that if c > 0 (a = 1), then the factorization will be either of the form (\underline{\;\;\;\;} + \underline{\;\;\;\;})(\underline{\;\;\;\;} + \underline{\;\;\;\;}) or (\underline{\;\;\;\;} - \underline{\;\;\;\;})(\underline{\;\;\;\;} - \underline{\;\;\;\;}) (same signs). If c < 0 (a = 1), then use the form (\underline{\;\;\;\;} - \underline{\;\;\;\;})(\underline{\;\;\;\;} - \underline{\;\;\;\;}) (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) = x^2 + 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) = x^2 + 10x + 21. Observe that it also shows how to factor x^2 + 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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