# 6.4 Special products

 Page 1 / 5
By the end of this section, you will be able to:
• Square a binomial using the Binomial Squares Pattern
• Multiply conjugates using the Product of Conjugates Pattern
• Recognize and use the appropriate special product pattern

Before you get started, take this readiness quiz.

1. Simplify: ${9}^{2}$ ${\left(-9\right)}^{2}$ $\text{−}{9}^{2}.$
If you missed this problem, review [link] .

## Square a binomial using the binomial squares pattern

Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the binomial    twice and using the methods of the last section, there is less work to do if you learn to use a pattern.

$\begin{array}{}\\ \\ \text{Let’s start by looking at}\phantom{\rule{0.2em}{0ex}}{\left(x+9\right)}^{2}.\hfill & & & \\ \text{What does this mean?}\hfill & & & \phantom{\rule{4em}{0ex}}{\left(x+9\right)}^{2}\hfill \\ \text{It means to multiply}\phantom{\rule{0.2em}{0ex}}\left(x+9\right)\phantom{\rule{0.2em}{0ex}}\text{by itself.}\hfill & & & \phantom{\rule{4em}{0ex}}\left(x+9\right)\left(x+9\right)\hfill \\ \text{Then, using FOIL, we get:}\hfill & & & \phantom{\rule{4em}{0ex}}{x}^{2}+9x+9x+81\hfill \\ \text{Combining like terms gives:}\hfill & & & \phantom{\rule{4em}{0ex}}{x}^{2}+18x+81\hfill \\ \\ \\ \\ \text{Here’s another one:}\hfill & & & \phantom{\rule{4em}{0ex}}{\left(y-7\right)}^{2}\hfill \\ \text{Multiply}\phantom{\rule{0.2em}{0ex}}\left(y-7\right)\phantom{\rule{0.2em}{0ex}}\text{by itself.}\hfill & & & \phantom{\rule{4em}{0ex}}\left(y-7\right)\left(y-7\right)\hfill \\ \text{Using FOIL, we get:}\hfill & & & \phantom{\rule{4em}{0ex}}{y}^{2}-7y-7y+49\hfill \\ \text{And combining like terms:}\hfill & & & \phantom{\rule{4em}{0ex}}{y}^{2}-14y+49\hfill \\ \\ \\ \\ \text{And one more:}\hfill & & & \phantom{\rule{4em}{0ex}}{\left(2x+3\right)}^{2}\hfill \\ \text{Multiply.}\hfill & & & \phantom{\rule{4em}{0ex}}\left(2x+3\right)\left(2x+3\right)\hfill \\ \text{Use FOIL:}\hfill & & & \phantom{\rule{4em}{0ex}}4{x}^{2}+6x+6x+9\hfill \\ \text{Combine like terms.}\hfill & & & \phantom{\rule{4em}{0ex}}4{x}^{2}+12x+9\hfill \end{array}$

Look at these results. Do you see any patterns?

What about the number of terms? In each example we squared a binomial and the result was a trinomial    .

${\left(a+b\right)}^{2}=\text{____}+\text{____}+\text{____}$

Now look at the first term in each result. Where did it come from?

The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!

${\left(a+b\right)}^{2}={a}^{2}+\text{____}+\text{____}$

To get the first term of the product, square the first term .

Where did the last term come from? Look at the examples and find the pattern.

The last term is the product of the last terms, which is the square of the last term.

${\left(a+b\right)}^{2}=\text{____}+\text{____}+{b}^{2}$

To get the last term of the product, square the last term .

Finally, look at the middle term . Notice it came from adding the “outer” and the “inner” terms—which are both the same! So the middle term is double the product of the two terms of the binomial.

$\begin{array}{c}{\left(a+b\right)}^{2}=\text{____}+2ab+\text{____}\hfill \\ {\left(a-b\right)}^{2}=\text{____}-2ab+\text{____}\hfill \end{array}$

To get the middle term of the product, multiply the terms and double their product .

Putting it all together:

## Binomial squares pattern

If $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b$ are real numbers,

$\begin{array}{}\\ \\ {\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}\hfill \\ {\left(a-b\right)}^{2}={a}^{2}-2ab+{b}^{2}\hfill \end{array}$

To square a binomial:

• square the first term
• square the last term
• double their product

A number example helps verify the pattern.

$\begin{array}{cccc}& & & \phantom{\rule{4em}{0ex}}{\left(10+4\right)}^{2}\hfill \\ \text{Square the first term.}\hfill & & & \phantom{\rule{4em}{0ex}}{10}^{2}+\text{___}+\text{___}\hfill \\ \text{Square the last term.}\hfill & & & \phantom{\rule{4em}{0ex}}{10}^{2}+\text{___}+{4}^{2}\hfill \\ \text{Double their product.}\hfill & & & \phantom{\rule{4em}{0ex}}{10}^{2}+2·10·4+{4}^{2}\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}100+80+16\hfill \\ \text{Simplify.}\hfill & & & \phantom{\rule{4em}{0ex}}196\hfill \end{array}$

To multiply ${\left(10+4\right)}^{2}$ usually you’d follow the Order of Operations.

$\begin{array}{c}\hfill {\left(10+4\right)}^{2}\hfill \\ \hfill {\left(14\right)}^{2}\hfill \\ \hfill 196\hfill \end{array}$

The pattern works!

Multiply: ${\left(x+5\right)}^{2}.$

## Solution

 Square the first term. Square the last term. Double the product. Simplify.

Multiply: ${\left(x+9\right)}^{2}.$

${x}^{2}+18x+81$

Multiply: ${\left(y+11\right)}^{2}.$

${y}^{2}+22y+121$

Multiply: ${\left(y-3\right)}^{2}.$

## Solution

 Square the first term. Square the last term. Double the product. Simplify.

Multiply: ${\left(x-9\right)}^{2}.$

${x}^{2}-18x+81$

Multiply: ${\left(p-13\right)}^{2}.$

${p}^{2}-26p+169$

Multiply: ${\left(4x+6\right)}^{2}.$

## Solution

 Use the pattern. Simplify.

Multiply: ${\left(6x+3\right)}^{2}.$

$36{x}^{2}+36x+9$

Multiply: ${\left(4x+9\right)}^{2}.$

$16{x}^{2}+72x+81$

Multiply: ${\left(2x-3y\right)}^{2}.$

## Solution

 Use the pattern. Simplify.

Multiply: ${\left(2c-d\right)}^{2}.$

$4{c}^{2}-4cd+{d}^{2}$

Multiply: ${\left(4x-5y\right)}^{2}.$

$16{x}^{2}-40xy+25{y}^{2}$

At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
divide 3x⁴-4x³-3x-1 by x-3
how to multiply the monomial
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
how do u solve that question
Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
Brandon has a cup of quarters and dimes with a total of 5.55\$. The number of quarters is five less than three times the number of dimes
app is wrong how can 350 be divisible by 3.
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
I'm getting "math processing error" on math problems. Anyone know why?
Can you all help me I don't get any of this
4^×=9
Did anyone else have trouble getting in quiz link for linear inequalities?
operation of trinomial
y=2×+9