# 9.5: Factoring Quadratic Expressions

## 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 .

## 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 and :

- . See Examples 1-4.

Additional Examples:

*Factor.*

a. . Answer:

b. . Answer:

c. . Answer:

- . See Examples 5 and 6.

Additional Examples:

*Factor.*

a. . Answer:

b. . Answer:

c. . Answer:

- . See Examples 7-9.

Additional Examples:

*Factor.*

a. . Answer:

b. . Answer:

c. . Answer:

- . See Example 10.

Additional Examples:

*Factor*.

a. . Answer:

b. . Answer:

c. . Answer:

- Allow students to infer that if , then the factorization will be either of the form or (same signs). If , 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 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.*

.

Solution. The diagram shows that . Observe that it also shows how to factor .

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

## Error Troubleshooting

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