Factorizing Quadratic Equations

Intro

If we observe the quadratic expression

,

 

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

.

 

Hence, there are factors of , however, the techniques of factorization are different from those presented in the tutorial on factoring expressions.

 

This tutorial concentrates on methods to factor quadratic expressions of the form

x2 coefficient equal to 1

We found in  the tutorial on multiplying binomial expressions that

.

Hence if we start with an expression of the form , our goal is to find a and b such that a+b=c and ab=d. If we can find such a and b then the expression and we have succeeded in factorizing .  From this we can obtain the following set of steps to factor quadratic expressions with x2 coefficient equal to 1.

Example 1:

Factor .


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 , 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: 

 

 

         So a=4 ,b=3 satisfies a+b=7 and ab=12.

       Hence 

 

Step 3.  Verify

 

 


Example 2:

Factor .


Solution:

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

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:

 

 

 

         So a=-7,b=3 satisfies a+b=-4 and ab=-21.

       Hence

 

Step 3.  Verify

 

 

Practice

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

 

 

x2 coefficient not equal to 1

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

.

Hence if we start with an expression of the form , our goal is to find a,b,c and d such that ac=e, ad+bc=f and bd=g. If we can find such a,b,c and d then the expression and we have factored .  From this we can obtain the following set of steps for factoring quadratic expressions.

Example 1:

Factor .


Solution:

 

 

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

 

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

 

As , a and c are not

interchangeable and we repeat pairs such as (2,1) and (1,2)

 

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

 

(5,1)

 

As , if we interchange

a and c,we do not need to interchange b and d as

interchanging both creates an equivalent expression.

Also, as , 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 , multiply the expressions  

           together and determine if one of these is equal to

 

 

        

(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

 

 Solution: 

 

 

 

 

Example 2:

Factor .


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)

 

 

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

           expression , multiply the expressions  

           together and determine if one of these is equal to

 

        

(a,c)

(b,d)

(ax+b)(cx+d)

Product

(3,1)

(12,-1)

(3x+12)(x+-1)

2x2+9x+-12

(3,1)

(6,-2)

(3x+6)(x+-2)

2x2+0x+-12

(3,1)

(4,-3)

(3x+4)(1x+-3)

2x2+-5x+-12

(1,3)

(12,-1)

(x+12)(3x+-1)

2x2+35x+-12

(1,3)

(6,-2)

(x+6)(3x+-2)

2x2+16x+-12

(1,3)

(4,-3)

(x+4)(3x+-3)

2x2+9x+-12

(-3,-1)

(12,-1)

(-3x+12)(-1x+-1)

2x2+-9x+-12

(-3,-1)

(6,-2)

(-3x+6)(-x+-2)

2x2+0x+-12

(-3,-1)

(4,-3)

(-3x+4)(-1x+-3)

2x2+5x+-12

(-1,-3)

(12,-1)

(-x+12)(-3x+-1)

2x2+-23x+-12

(-1,-3)

(6,-2)

(-x+6)(-3x+-2)

2x2+-16x+-12

(-1,-3)

(4,-3)

(-x+4)(-3x+-3)

2x2-9x+-12

 

 Solution: 

 

 

 

 

Practice

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