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Basic Quadratic Factoring

Basic Quadratic Factoring

Author: Sophia Tutorial
Description:

Factor a quadratic expression in the form x²+bx+c.

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Tutorial
what's covered
  1. Binomial Multiplication
  2. Basic Quadratic Factoring
  3. The Sign of pq

1. Binomial Multiplication

When factoring quadratic expressions, we are rewriting a quadratic from the form a x squared plus b x plus c into the form left parenthesis a x plus p right parenthesis left parenthesis b x plus q right parenthesis (Note: the variables a and b are not necessarily equivalent in both forms: they just represent a variable number).

To better understand what we are looking for when we are factoring a quadratic, it is helpful to look at how we go in the opposite direction. This process is known as binomial multiplication, and is modeled by FOIL: First, Outside, Inside, Last.

open parentheses x plus 2 close parentheses open parentheses x minus 3 close parentheses

x squared
Multiply first terms: x times x
x squared minus 3 x
Multiply outside terms: x times short dash 3
x squared minus 3 x plus 2 x
Multiply inside terms: 2 times x
x squared minus 3 x plus 2 x minus 6
Multiply last terms: 2 times short dash 3
x squared minus x minus 6
Combine like terms

big idea
Look at the coefficient of the x-term, and the constant term. When factoring a quadratic expression, we are looking for two integers, p and q. The sum of p and q is the coefficient of the x-term, and the product of p and q is the constant term.
In the example above, we can factor x squared minus x minus 6 as left parenthesis x plus 2 right parenthesis left parenthesis x minus 3 right parenthesis. We know this is true, because we FOILed left parenthesis x plus 2 right parenthesis left parenthesis x minus 3 right parenthesis to get x squared minus x minus 6. Thinking about the relationship between factoring and FOIL, we see that the sum of 2 and -3 is -1 (the coefficient of x), and the product of 2 and -3 is -6 (the constant term).


2. Basic Quadratic Factoring

When factoring a quadratic, we need to identify two integers whose product is the constant term of the quadratic, and whose sum is the x-term coefficient. To identify these two integers, we identify pairs of factors that multiply to the constant term. From there, we add the integers in each pair together, and work with the pair that sums to the x-term coefficient.

Factor x squared plus 8 x plus 15

To begin, we will list pairs of integers that multiply to 15:

p q
1 15
-1 -15
3 5
-3 -5

Note that it can include two negative numbers, because the product will be positive. Next, we add p and q, looking for a sum of 8. Once we find our sum of 8, there is no point in finding the other sums, because we know we don't want to use those values for p and q.

p
q
sum
1 + 15 equals 16
-1 + -15 equals -16
3 + 5 equals 8
-3 + -5 equals


We have identified p as 3, and q as 5. This means that we can factor the quadratic as:

x squared plus 8 x plus 15 space equals space open parentheses x plus 3 close parentheses open parentheses x plus 5 close parentheses


3. The Sign of pq

Looking at the sign of the constant term can help you eliminate possible values for p and q when factoring. For example, if the constant term is negative, we know that one of p or q must be negative, but not both (because two negative values result in a positive number when multiplied). The table below can guide you when thinking about the sign of pq:


q -q
p pq -pq
-p -pq pq

Here is how we can use the sign of pq to narrow our focus when factoring:

EXAMPLE

Factor x squared minus 7 x plus 12

We see that the constant term is positive. This means that the signs of p and q must match: either both of them are positive numbers, or both of them are negative. Next, we look at the x-term. The coefficient is a negative number. This means that the sum of p and q must be negative. Since the sum of two positive numbers is always positive, we can conclude that p and q must be negative.

So we draft up pairs of negative numbers that equal positive 12 when multiplied:

p q
-1 -12
-2 -6
-3 -4

hint
Note that there is no need to write the other pairs of p and q that essentially switch what we already have. Their sums will be the same, because addition is commutative.

Now we add our integer pairs, and stop when we reach –7:

p
q
sum
short dash 1 + open parentheses short dash 12 close parentheses equals short dash 13
short dash 2 + open parentheses short dash 6 close parentheses equals short dash 8
short dash 3 + open parentheses short dash 4 close parentheses equals short dash 7

This means we can factor as:

x squared minus 7 x plus 12 space equals space open parentheses 3 minus x close parentheses open parentheses x minus 4 close parentheses

EXAMPLE

Factor x squared minus 4 x minus 12

We see that the constant term is negative. This means that one of p or q must be negative, but not both. Next, we look at the x-term. The coefficient is a negative number. This means that the larger number will be negative and the smaller number will be positive.

So we draft up pairs of one positive number and one negative number that equals negative 12 when multiplied:

p q
-12 1
-6 2
-4 3

Notice in each case, the larger number had the negative sign (12 is larger than 1, so the negative sign will go with the 12). Now we add our integer pairs, and stop when we reach a sum of –4:

p
q
sum
short dash 12 + 1 equals short dash 11
short dash 6 + 2 equals short dash 4
short dash 4 + 3 equals

This means we can factor as: x squared minus 4 x minus 12 space equals space left parenthesis x minus 6 right parenthesis left parenthesis x plus 2 right parenthesis

summary
With binomial multiplication, we use the FOIL method. Basic quadratic factoring is the process of writing an equation from expanded form to factored form. Factoring involves finding two integers whose sum is the coefficient of the x-term, and whose product is the constant term of the quadratic equation. The sign of pq can help determine the two integers.
Formulas to Know
Basic Quadratic Factoring

open parentheses x plus p close parentheses open parentheses x plus q close parentheses equals x squared plus left parenthesis p plus q right parenthesis x plus p q