Factoring Quadratic Expressions ax2 + bx + c

Factoring Quadratic Expressions ax2 bx c

Setting the Stage Do you remember how to multiply these together? (Referred to as FOIL in some books) (2x 3)(3x – 4) 6x2 – 8x 9x – 12 6x2 x – 12 How about another way? NEXT 2x 3 3x 6x2 9x –4 – 8x – 12 Complete the inside of the box by multiplying the numbers outside the box that are in the same row and column. : If you combine all the like terms in the box you get the product of (2x 3) and (3x – 4)

What is Factoring? Basically, we are going to undo FOIL. Starting with a quadratic expression like the one below, 6x2 x – 12 we are going to find two binomials that when multiplied together give us the original expression. In other words, we are going to start with the expression above and try to find the expression (2x 3)(3x – 4) NEXT

How To Do It Let’s start with 6x2 x – 12 Make a box and put the first term in the top left and last term in the bottom right. 6x 2 9x – 8x – 12 Put these two numbers in the box . These last two numbers are used to get the middle term in our original expression, so they both need a x. NEXT To find the other two numbers, here’s what you need to do Make a table on the left side put the product of a and c, and on the right side put b ac b – 72 1 (– 8)(9) (– 8) (9) Now, find TWO numbers that multiply to make – 72 and add to make 1. Click the screen when you think you have them.

How to Do It Once all the numbers are in the box, we need to determine which numbers go on the outside of the box. Instead of just staring at the box until you magically know what to do, start by finding the GCF of the first row. GCF 3x ? – ?4 2x 6x2 – 8x ?3 9x – 12 (2x)( ? ) 6x2 (2x)( ? ) – 8x (3x)( ? ) 9x Our original expression can now be factored using the numbers outside the box. 6x2 x – 12 (3x – 4)(2x 3) NEXT

Practice Time Try each of these on your own when finished, click on the “answer” box to see if you are correct. If you get it wrong, or you get stuck, click on the “step by step solution” box. #1: 3x2 – 14x – 5 ANSWER STEP BY STEP SOLUTION #2: x – 12x 32 2 ANSWER STEP BY STEP SOLUTION #3: 20x 23x 6 2 ANSWER STEP BY STEP SOLUTION

Answer to Practice #1 (3x 1)(x – 5) Back to Practice

Answer to Practice #2 (x – 8)(x – 4) Back to Practice

Answer to Practice #3 (4x 3)(5x 2) Back to Practice

Step by Step Solution to Practice #1 After the directions show up on the right, click the screen to complete each step. Try to complete the next step on your own before clicking the screen. Click the screen to begin. x –5 3x 1 3x2 1x –15x –5 ac b – 15 – 14 (– 15)(1) (– 15) (1) Put the first term and the last term in the box Make a chart with ac on the left and b on the right Find TWO numbers that multiply to make the number on the left and add to make the number on the right. Put these two numbers in the box. (Be sure they both have an x) Find the numbers on the outside of the box. (use the GCF of the first row) Write your answer in factored form. 3x2 – 14x – 5 (3x 1)(x – 5) DONE! Back to Practice

Step by Step Solution to Practice #2 After the directions show up on the right, click the screen to complete each step. Try to complete the next step on your own before clicking the screen. Click the screen to begin. x –4 x –8 x2 – 8x –4x 32 ac b 32 – 12 (– 8)(– 4) (– 8) (– 4) Put the first term and the last term in the box Make a chart with ac on the left and b on the right Find TWO numbers that multiply to make the number on the left and add to make the number on the right. Put these two numbers in the box. (Be sure they both have an x) Find the numbers on the outside of the box. (use the GCF of the first row) Write your answer in factored form. x2 – 12x 32 (x – 8)(x – 4) DONE! Back to Practice

Step by Step Solution to Practice #3 After the directions show up on the right, click the screen to complete each step. Try to complete the next step on your own before clicking the screen. Click the screen to begin. 4x 5x 2 20x2 3 15x 8x 6 ac b 120 23 (8)(15) (8) (15) Put the first term and the last term in the box Make a chart with ac on the left and b on the right Find TWO numbers that multiply to make the number on the left and add to make the number on the right. Put these two numbers in the box. (Be sure they both have an x) Find the numbers on the outside of the box. (use the GCF of the first row) Write your answer in factored form. 20x2 23x 6 (4x 3)(5x 2) DONE! Back to Practice