# Factoring Quadratic Equations

### Objectives

This lesson presents the necessary background material to:

• Factor expressions of the form ${x}^{2}+\mathrm{bx}+c$
• Factor expressions of the form ${\mathrm{ax}}^{2}+\mathrm{bx}+c$

### Introduction

If we observe the quadratic expression ${x}^{2}+3x+2$ we can readily see that there is no common factor to the three terms. However, in the tutorial on multiplying binomial expressions we learned that

$\left(x+1\right)\left(x+2\right)={x}^{2}+3x+2$
Hence there are factors of ${x}^{2}+3x+2$, however we will need new techniques for factoring such expressions. This lesson lesson will present the new methods we need to factor expressions of the form

${\mathrm{ax}}^{2}+\mathrm{bx}+c$

### Factoring x2 + bx + c

We found in the tutorial on multiplying binomial expressions that

$\left(x+a\right)\left(x+b\right)={x}^{2}+\left(a+b\right)x+\mathrm{ab}$
Hence if we start with an expression of the form ${x}^{2}+\mathrm{cx}+d$, our goal is to find $a$ and $b$ such that $a+b=c$ and $\mathrm{ab}=d$. If we can find such $a$ and $b$ then the expression

${x}^{2}+\mathrm{cx}+d={x}^{2}+\left(a+b\right)x+\mathrm{ab}=\left(x+a\right)\left(x+b\right)$
and we have succeeded in factoring ${x}^{2}+\mathrm{cx}+d$. From this we can obtain the following set of steps to factor quadratic expressions with x2coefficient equal to 1.

### Example 1:

Factor: ${x}^{2}+7x+12$

### Solution:

 Step 1. Find all pairs of (a, b) such that ab = 12: (12,1), (6,2), (4,3), (-12,-1), (-6,-2), (-4,-3) As: $\left(x+a\right)\left(x+b\right)=\left(x+b\right)\left(x+a\right)$ $a$ and $b$ are interchangeable and we do not repeat pairs such as (12,1) and (1,12). Step 2. Determine if one of these pairs satisfies a + b = 7:   $12+1\ne 7;6+2\ne 7;4+3=7$ So a= 4, b= 3 satisfies a + b = 7 and ab = 12. Hence: ${x}^{2}+7x+12=\left(x+4\right)\left(x+3\right)$ Step 3. Verify   $\left(x+4\right)\left(x+3\right)={x}^{2}+3x+4x+12$ $\left(x+4\right)\left(x+3\right)={x}^{2}+7x+12$

### Example 2:

Factor: ${x}^{2}-4x-21$

### Solution:

Note: For convenience of our method we will rewrite differences as sums with negative numbers so the problem becomes factor

${x}^{2}+\left(-4x\right)+\left(-21\right)$

 Step 1. Find all pairs of (a,b) such that ab = -21: (21,-1), (-21,1), (7,-3), (-7,3) Step 2. Determine if one of these pairs satisfies               a + b = -4: $21+\left(-1\right)\ne -4;-21+1\ne -4;$ $7+\left(-3\right)\ne -4;-7+3=-4.$               So a= -7, b= 3 satisfies a + b = -4 and ab = -21.               Hence: ${x}^{2}-4x-21=\left(x+\left(-7\right)\right)\left(x+3\right)$ Step 3. Verify   $\left(x+\left(-7\right)\right)\left(x+3\right)={x}^{2}+3x+\left(-7x\right)+\left(-21\right)$ $\left(x-7\right)\left(x+3\right)={x}^{2}-4x-21$

### Practice 1.

Click on the link below to practice factoring quadratic expressions with leading coefficient equal to 1.

### Factoring ax2 + bx + c

We can multiply the following expressions together to yield the following::

$\left(\mathrm{ax}+b\right)\left(\mathrm{cx}+d\right)={\mathrm{acx}}^{2}+\left(\mathrm{ad}+\mathrm{cb}\right)x+\mathrm{bd}$
Hence if we start with an expression of the form ${\mathrm{ex}}^{2}+\mathrm{fx}+g$, our goal is to find $a$,$b$, $c$ and $d$ such that $\mathrm{ac}=e$, $\mathrm{ad}+\mathrm{bc}=f$ and $\mathrm{bd}=g$. If we can find such $a$,$b$, $c$ y $d$ then the expression

$\left(\mathrm{ax}+b\right)\left(\mathrm{cx}+d\right)={\mathrm{acx}}^{2}+\left(\mathrm{ab}+\mathrm{cd}\right)x+\mathrm{bd}={\mathrm{ex}}^{2}+\mathrm{fx}+g$
and we have factored: ${\mathrm{ex}}^{2}+\mathrm{fx}+g$. From this we can obtain the following set of steps for factoring quadratic expressions.

### Example 1:

Factor: ${\mathrm{2x}}^{2}+11x+5$

### Solution:

Step 1. Find all pairs of (a,c) such that ac = 2:

(2,1), (1,2), (-2,-1), (-1,-2)

As: $\left(\mathrm{ax}+b\right)\left(\mathrm{cx}+d\right)\ne \left(\mathrm{cx}+b\right)\left(\mathrm{ax}+d\right)$
$a$ and $c$ nare not interchangeable and we repeat pairs such as (2,1) y (1,2).

Step 2. Find all pairs of (b,d) such that bd = 5:

$\left(5,1\right)$
As: $\left(\mathrm{ax}+b\right)\left(\mathrm{cx}+d\right)=\left(\mathrm{cx}+b\right)\left(\mathrm{ax}+d\right)$
if we interchange $a$ and $c$, we do not need to interchange $b$ and $d$ as interchanging both creates an equivalent expression.

Also, as : $\left(\mathrm{ax}+b\right)\left(\mathrm{cx}+d\right)=\left(-\mathrm{ax}+\left(-b\right)\right)\left(-\mathrm{cx}+\left(-d\right)\right)$
we do not need to include both (x, y) and (-x,-y). One is enough.

Step 3. Substitute all combinations of (a,c) and (b,d) into the expression :

$\left(\mathrm{ax}+b\right)\left(\mathrm{cx}+d\right)$
multiply the expressions together and determine if one of these is equal to :

${\mathrm{2x}}^{2}+11x+5$

 (a,c) (b,d) (ax+b)(cx+d) Product (2,1) (5,1) (2x+5)(1x+1) 2x2+7x+5 (1,2) (5,1) (x+5)(2x+1) 2x2+11x+5 (-2,-1) (5,1) (-2x+5)(-1x+1) 2x2-7x+5 (-1,-2) (5,1) (-x+5)(-2x+1) 2x2-11x+5

### Example 2:

Factor: ${\mathrm{3x}}^{2}+5x-12$

### Solution:

Step 1. Find all pairs of (a,c) such that ac = 3:

(3,1), (1,3), (-3,-1), (-1,-3)

Step 2. Find all pairs of (b,d) such that bd = -12:

(12,-1), (6,-2), (4,-3)

Paso 3: Substitute all combinations of (a,c) and (b,d) into the expression:

$\left(\mathrm{ax}+b\right)\left(\mathrm{cx}+d\right)$
multiply the expressions together and determine if one of these is equal to :

${\mathrm{3x}}^{2}+5x-12$

 (a,c) (b,d) (ax+b)(cx+d) Product (3,1) (12,-1) (3x+12)(x-1) 3x2+9x -12 (3,1) (6,-2) (3x+6)(x-2) 3x2+0x-12 (3,1) (4,-3) (3x+4)(1x-3) 3x2-5x -12 (1,3) (12,-1) (x+12)(3x-1) 3x2+35x -12 (1,3) (6,-2) (x+6)(3x-2) 3x2+16x-12 (1,3) (4,-3) (x+4)(3x-3) 3x2+9x-12 (-3,-1) (12,-1) (-3x+12)(-1x-1) 3x2-9x-12 (-3,-1) (6,-2) (-3x+6)(-x-2) 3x2+0x-12 (-3,-1) (4,-3) (-3x+4)(-1x-3) 3x2+5x-12 (-1,-3) (12,-1) (-x+12)(-3x-1) 3x2-23x-12 (-1,-3) (6,-2) (-x+6)(-3x-2) 3x2-16x-12 (-1,-3) (4,-3) (-x+4)(-3x-3) 3x2-9x-12

### Practice 2.

Click on the link below to practice factoring quadratic expressions with leading coefficient not equal to 1.

### Summary

Now that you've completed this lesson, you should be able to:

• Factor expressions of the form ${x}^{2}+\mathrm{bx}+c$
• Factor expressions of the form ${\mathrm{ax}}^{2}+\mathrm{bx}+c$