Factoring Quadratic Expressions ax 2 + bx + c

2x2x 3x3x +3 – 4 Setting the Stage Do you remember how to multiply these together? (Referred to as FOIL in some books) How about another way? Complete the inside of the box by multiplying the numbers outside the box that are in the same row and column. NEXT ♪ : If you combine all the like terms in the box you get the product of (2x + 3) and (3x – 4) (2x + 3)(3x – 4) = 6x 2 – 8x + 9x – 12 = 6x26x2 – x – 8x 6x 2 + x – 12

What is Factoring? Basically, we are going to undo FOIL. Starting with a quadratic expression like the one below, 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 NEXT 6x 2 + x – 12 (2x + 3)(3x – 4)

How To Do It … Let’s start with Make a box and put the first term in the top left and last term in the bottom right. To find the other two numbers, here’s what you need to do … – 72 1 (– 8)(9) (– 8) + (9) 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. Make a table … on the left side put the product of a and c, and on the right side put b ac b Now, find TWO numbers that multiply to make – 72 and add to make + 1. Click the screen when you think you have them. NEXT 6x 2 + x – 12 6x26x2 – x – 8x

? ? 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. 3x3x 2x2x 3 – 4 Our original expression can now be factored using the numbers outside the box. NEXT 6x26x2 – x – 8x GCF = (2x)( ? ) = 6x 2 (2x)( ? ) = – 8x ? (3x)( ? ) = + 9x 6x 2 + x – 12 = (3x – 4)(2x + 3)

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: ANSWER STEP BY STEP SOLUTION 3x 2 – 14x – 5 #2: ANSWER STEP BY STEP SOLUTION x 2 – 12x + 32 #3: ANSWER STEP BY STEP SOLUTION 20x x + 6

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

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

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

Step by Step Solution to Practice #1 Back to Practice ac b – 15 – 14 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. (– 15)(1) (– 15) + (1) 3x3x x 1 – 5 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. DONE! 3x23x2 1x1x –5 –15x 3x 2 – 14x – 5 = (3x + 1)(x – 5)

Step by Step Solution to Practice #2 Back to Practice ac b 32 – 12 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. x x – 8 – 4 (– 8)(– 4) (– 8) + (– 4) 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. DONE! x2x2 – 8x +32 –4x x 2 – 12x + 32 = (x – 8)(x – 4)

Step by Step Solution to Practice #3 Back to Practice ac b 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. (8)(15) 4x4x 5x 3 2 (8) + (15) DONE! 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. 20x 2 15x 8x8x+ 6 20x x + 6 = (4x + 3)(5x + 2)