9.5: Factoring Quadratic Expressions
In this lesson, we will learn how to factor quadratic polynomials for different values of
Factoring Quadratic Expressions in Standard From
Quadratic polynomials are polynomials of degree 2. The standard form of a quadratic polynomial is
Example 1: Factor
Solution: We are looking for an answer that is a product of two binomials in parentheses:
To fill in the blanks, we want two numbers
6&=2 \times 3 \qquad and \qquad 2+3=5
So the answer is
We can check to see if this is correct by multiplying
2 is multiplied by
Combine the like terms:
Example 2: Factor
Solution: We are looking for an answer that is a product of the two parentheses
The number 8 can be written as the product of the following numbers.
8&=2 \times 4 && and && 2+4=6
And
The answer is
Example 3: Factor
Solution: We are looking for an answer that is a product of two parentheses
In this case, we must take the negative sign into account. The number –15 can be written as the product of the following numbers.
And also,
The answer is
Example 4: Factor
Solution: First factor the common factor of –1 from each term in the trinomial. Factoring –1 changes the signs of each term in the expression.
We are looking for an answer that is a product of two parentheses
Now our job is to factor
The number –6 can be written as the product of the following numbers.
&6=1 \times (6) \qquad and \qquad 1+(6)=5\\
&6=(2) \times 3 \qquad and \qquad (2)+3=1\\
&6=2 \times (3) \qquad and \qquad 2+(3)=1 \qquad This \ is \ the \ correct \ choice.
The answer is
To Summarize:
A quadratic of the form
 If
b andc are positive then bothm andn are positive. Example
x2+8x+12 factors as(x+6)(x+2) .
 Example
 If
b is negative andc is positive then bothm andn are negative. Example
x2−6x+8 factors as(x−2)(x−4) .
 Example
 If
c is negative then eitherm is positive andn is negative or viceversa. Example
x2+2x−15 factors as(x+5)(x−3) .  Example
x2+34x−35 factors as(x+35)(x−1) .
 Example
 If
a=−1 , factor a common factor of –1 from each term in the trinomial and then factor as usual. The answer will have the form−(x+m)(x+n) . Example
−x2+x+6 factors as−(x−3)(x+2) .
 Example
Practice Set
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both.
CK12 Basic Algebra: Factoring Quadratic Equations (16:30)
Factor the following quadratic polynomials.

x2+10x+9 
x2+15x+50 
x2+10x+21 
x2+16x+48 
x2−11x+24 
x2−13x+42 
x2−14x+33 
x2−9x+20 
x2+5x−14 
x2+6x−27 
x2+7x−78 
x2+4x−32 
x2−12x−45 
x2−5x−50 
x2−3x−40 
x2−x−56 
−x2−2x−1 
−x2−5x+24 
−x2+18x−72 
−x2+25x−150 
x2+21x+108 
−x2+11x−30 
x2+12x−64 
x2−17x−60
Mixed Review
 Evaluate
f(2) whenf(x)=12x2−6x+4 .  The Nebraska Department of Roads collected the following data regarding mobile phone distractions in traffic crashes by teen drivers.
 Plot the data as a scatter plot.
 Fit a line to this data.
 Predict the number of teenage traffic accidents attributable to cell phones in the year 2012.
Year ( 
Total ( 

2002  41 
2003  43 
2004  47 
2005  38 
2006  36 
2007  40 
2008  42 
2009  42 
 Simplify
405−−−√ .  Graph the following on a number line:
−π,2√,53,−310,16−−√ .  What is the multiplicative inverse of
94 ?
Quick Quiz
 Name the following polynomial. State its degree and leading coefficient
6x2y4z+6x6−2y5+11xyz4 .  Simplify
(a2b2c+11abc5)+(4abc5−3a2b2c+9abc) .  A rectangular solid has dimensions
(a+2) by(a+4) by(3a) . Find its volume.  Simplify
−3hjk3(h2j4k+6hk2) .  Find the solutions to
(x−3)(x+4)(2x−1)=0 .  Multiply
(a−9b)(a+9b) .