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Difficulty Level: At Grade 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:

## 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.

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.

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.

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.

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.

Feb 22, 2012

Aug 22, 2014

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