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Intermediate Algebra – Student Workbook – Second Edition 2013

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Lesson 5b - Solving Quadratic Equations<br />

Mini-Lesson<br />

Factoring Quadratic Expressions and Solving Quadratic Equations by Factoring<br />

(Factoring using Greatest Common Factor Method)<br />

Let’s use the information on the previous page to help us FACTOR 3x 2 + 6x. The building<br />

blocks of 3x 2 + 6x are the terms 3x 2 and 6x. Each is written in FACTORED FORM below.<br />

3x 2 = 3 * x * x<br />

and 6x = 3 * 2 * x<br />

Let’s rearrange these factorizations just slightly as follows:<br />

×<br />

3x 2 = (3 * x) * x and 6x = (3 * x) * 2<br />

We can see that (3 x) = 3x is a common FACTOR to both terms. In fact, 3x is the<br />

GREATEST COMMON FACTOR to both terms.<br />

Let’s rewrite the full expression with the terms in factored form and see how that helps us factor<br />

the expression:<br />

3x 2 + 6x = (3 * x) * x + (3 * x) * 2<br />

= (3x) * x + (3x) * 2<br />

= (3x)(x + 2)<br />

Always check your factorization by multiplying the final result. (3x)(x + 2)=3x 2 + 6x CHECKS<br />

3x 2 + 6x = (3x)(x + 2) is in COMPLETELY FACTORED FORM<br />

How can we use this factored form to solve quadratic equations such as 3x 2 + 6x = 0?<br />

To solve 3x 2 + 6x = 0, FACTOR the left<br />

side:<br />

(3x)(x + 2) = 0<br />

Set each linear factor to 0:<br />

3x = 0 OR x + 2 = 0<br />

Solve each linear equation to get<br />

x = 0 OR x = -2<br />

The solutions to 3x 2 + 6x = 0 are<br />

x = 0 or x = -2.<br />

If you graph the parabola 3x 2 + 6x, I see that<br />

it crosses the x-axis at (0, 0) and (-2, 0)<br />

which confirms the solutions found above.<br />

Scottsdale Community College Page 213 <strong>Intermediate</strong> <strong>Algebra</strong>

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