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Squaring a Binomial Using the Binomial Squares Pattern

Squaring 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?

This figure has three columns. The first column contains the expression x plus 9, in parentheses, squared. Below this is the product of x plus 9 and x plus 9. Below this is x squared plus 9x plus 9x plus 81. Below this is x squared plus 18x plus 81. The second column contains the expression y minus 7, in parentheses, squared. Below this is the product of y minus 7 and y minus 7. Below this is y squared minus 7y minus 7y plus 49. Below this is the expression y squared minus 14y plus 49. The third column contains the expression 2x plus 3, in parentheses, squared. Below this is the product of 2x plus 3 and 2x plus 3. Below this is 4x squared plus 6x plus 6x plus 9. Below this is 4x squared plus 12x plus 9.

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,

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

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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!

Example

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

Solution

 x plus 5, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.
Square the first term.x squared plus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.
Square the last term.x squared plus blank plus 5 squared.
Double the product.x squared plus 2 times x times 5 plus 5 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.
Simplify.x squared plus 10 x plus 25.

Example

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

Solution

 y minus 3, in parentheses, squared. Above the expression is the general formula a minus b, in parentheses, squared.
Square the first term.y squared minus blank plus blank. Above the expression is the general form a squared plus 2 a b plus b squared.
Square the last term.y squared minus blank plus 3 squared.
Double the product.y squared minus y times y times 3 plus 3 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.
Simplify.y squared minus 6 y plus 9.

Example

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

Solution

 4 x plus 6, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.
Use the pattern.4 x squared plus 2 times 4 x times 6 plus 6 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.
Simplify.16 x squared plus 48 x plus 36.

Example

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

Solution

 contains 2 x minus 3 y, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.
Use the pattern.2 x squared minus 2 times 2 x times 3 y plus 3 y squared. Above this expression is the general formula a squared minus 2 times a times b plus b squared.
Simplify.4 x squared minus 12 x y plus 9 y squared.

Example

Multiply: \({\left(4{u}^{3}+1\right)}^{2}.\)

Solution

 4 u cubed plus 1, in parentheses, squared. Above the expression is the general formula a plus b, in parentheses, squared.
Use the pattern.4 u cubed, in parentheses, squared, plus 2 times 4 u cubed times 1 plus 1 squared. Above this expression is the general formula a squared plus 2 times a times b plus b squared.
Simplify.16 u to the sixth power plus 18 u cubed plus 1.

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