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Solving Equations by Factoring

Using Factoring to Solve Equations

How is factoring useful in solving equations? Consider, for example the factored equation

(3x - 2)( x + 1) = 0.

To solve this, use the following important principle.

Zero Product Property

For all numbers a and b, if ab = 0, then a = 0, b = 0, or both a and b equal 0.

Since (3x - 2)( x + 1) = 0, then the Zero Product Property states that either 3x - 2 = 0, x + 1 = 0, or both equal zero.

3x - 2 = 0 or x + 1 = 0
3x = 2   x = -1
   

So, the solutions of (3x - 2)( x + 1) = 0 are and x = -1. Now apply this property to solve an equation that must be factored first.

 

Example 1

Find the solutions of 2x 3 -2x 2 = 12x .

Solution

First, write this equation so that it is set equal to zero.

2x 3 - 2x 2 - 12x = 0  
x ( 2x 2 - 2x - 12 ) = 0 Since x is a factor of all the terms, factor it out.
2x ( x 2 - x - 6 ) = 0 Since each coefficient is even, factor out a 2.

Now factor x 2 - x - 6 as a product ( x + a )( x + b ) for some values of a and b.

( x + a )( x + b ) = x 2 + ( a + b )x + ab
  = x 2 - x - 6

In other words, a + b = -1 and ab = -6. By looking at the factors of -6, we see that a = 2 and b = -3 satisfy these requirements. So x 2 - x - 6 = ( x + 2)( x - 3). (Check to see if this is correct by multiplying it out.)

2x ( x 2 - x - 6 ) = 0  
2x( x + 2)( x - 3) = 0 Substitution

Use the Zero Product Property to find the solutions.

2x = 0 x + 2 = 0 x - 3 = 0
x = 0 x = -2 x = 3

So, the solutions of the equation are x = 0, x = -2, and x = 3.

Finally, point out that the solutions of the equation are the roots (zeros) of the function

y = 2x 3 - 2x 2 - 12x.

Graphically, the roots are where the graph of the function y = 2x 3 - 2x 2 - 12x intersects the x -axis. Plot some points or use a graphing calculator to graph this function.

The roots are at x = 0, x = -2, and x = 3, as shown on the figuer below.

 
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