# Factoring Quadratic Equations (a = 1 Case)

Topic: Factoring Quadratic Equations When a = 1

Introduction

Another factoring method for quadratic equations will be shown here. We deal with the case of in . If then another method may be needed. (Also if then we no longer have a quadratic equation.)

It is assumed that the reader is familiar with common factoring.

The Factoring Method

To illustrate the method, an example will be used.

Suppose that we are given . We need two numbers and which are factors of 4 and satisfy = 4 and . The two numbers which fit that criteria are 1 and 4 (or 4 and 1) since and .

Using the numbers 1 and 4 we can factor into . The equation follows the factored form format of .

To check that is indeed the factored form of , we use the FOIL method when multiplying binomials.

There are times when . As an example, the factored form of is .

The General Method

Given a quadratic equation of the form , we can factor it into the form . To determine what and , we seek two numbers and such that (since here and .

The examples below will illustrate how the general method works.

Examples

Example One

Factor .

Two factors of 12 which multiply to 12 are 1 and 12, 2 and 6, 3 and 4 (you can inculde the reverse ordered pairs too). Out of the pairs which multiply to 12, 3 and 4 sum to 7. The quadratic equation would be factored as .

Example Two

Factor .

Factors of 99 are 1 and 99, 3 and 33, 9 and 11. The two numbers which multiply together to get 99 and add together to get 20 are 9 and 11. The factored form of would be .

Example Three

Factor by factoring out as a common factor first.

From factoring out the (-1), we have a factorable quadratic in the brackets. The only factors of 2 are 1 and 2. We can factor  into .

Example Four

Factor .

In this equation we have , and in . Unlike examples one and two, we have to be careful with signs. I prefer to ignore the negative and look at the factors of .

Factors of 6 are 1 and 6, 2 and 3. Now, we consider the signs. One of the two numbers in the factor pair is negative. The pair 6 and 1 would not work as 6 – 1 = 5 and 1 – 6 = -5. With 2 and 3, we have 2 – 3 = -1 and 3 – 2 = 1. We go with 2 and -3.

The factored form of is .

Practice Problems

Here are some practice problems to build understanding.

1) Factor .

2) Factor .

3) Factor .

4) Factor .

5) Factor .

6) In , factor a (-1) first and then factor using the method described in this page.

7) Using common factoring first, factor .

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