Introduction

When we want to calculate the square of a binomial, like

(a+b)^2,

there's a formula that we can use instead of working out the product (a+b)(a+b) manually. The formula is

(a + b)^2 = a^2 + 2ab + b^2.

It might be helpful to remember this as "the square of the first term, plus the square of the second term, plus twice the product."

Suppose we need to calculate the square (x+3)^2. In this case, we have a= x and b=3. We substitute into the formula and simplify:

\begin{align} (x+3)^2 & = \\[3pt] (x)^2 + 2 (x) (3) + (3)^2 & = \\[3pt] x^2 +6 x + 9 & \end{align}

We can check that the formula gives the correct result by working out the multiplication manually:

\eqalign{ (x + 3)^2 &= \\[3pt] (x + 3)(x + 3) &= \\[3pt] x(x + 3) + 3(x + 3) &= \\[3pt] x^2 + 3x + 3x + 9 &= \\[3pt] x^2 + 6x + 9 &\, {\color{green}{\checkmark}} }

So we get the same result! But by using the formula, we save a lot of time and effort.

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Example: Squaring Linear Binomials

Expand (4+3y)^2.

EXPLANATION

We will use the following formula:

(a+b)^2 = a^2 + 2ab + b^2

Substituting a=4 and b=3y, we have

\eqalign{ (4 + 3y)^2 &= \\ (4)^2 + 2(4) ( 3y) + (3y)^2 &= \\ 16 + 24y + 9y^2. }

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Practice: Squaring Linear Binomials

The expression $(x + 4)^2$ is equivalent to

a
 $x^2+8x+16$
b
 $x^2+4x+4$
c
 $x^2+8x$
d
 $x^2+16$
e
 $2(x+4)$
Practice: Squaring Linear Binomials

$\left(7q + 3\right)^2 =$

a
b
c
d
e
Practice: Squaring Linear Binomials

$\left(\dfrac{1}{2}+3y\right)^2 =$

a
 $\dfrac{1}{4}+9y^2$
b
 $\dfrac{1}{4}+\dfrac{3}{2}y+9y^2$
c
 $\dfrac{1}{2}+6y+3y^2$
d
 $\dfrac{1}{2}+3y+3y^2$
e
 $\dfrac{1}{4}+3y+9y^2$
Example: Squaring Nonlinear Binomials

Expand (2n^2 + 3)^2.

EXPLANATION

We will use the following formula:

(a+b)^2 = a^2 + 2ab + b^2

Substituting a=2n^2 and b=3, we have

\eqalign{ (2n^2 + 3)^2 &= \\[2pt] (2n^2)^2 + 2(2n^2) ( 3) + (3)^2 &= \\[2pt] 2^2\cdot (n^2)^2 + 2(2n^2) (3) + (3)^2 &= \\[2pt] 4n^4 + 12n^2 + 9. }

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Practice: Squaring Nonlinear Binomials

$\left(x^2 + 5\right)^2 \:\!= \:\: $

a
b
c
d
e
Practice: Squaring Nonlinear Binomials

$\left(a^2+b^2\right)^2=$

a
 $a^4+2a^2b^2+b^4$
b
 $2\left(a^2+b^2\right)$
c
 $a^4+a^2b^2+b^4$
d
 $a^2+2a^2b^2+b^2$
e
 $a^2+2ab+b^2$
Practice: Squaring Nonlinear Binomials

$\left(2x^4 + y^3\right)^2 =$

a
b
c
d
e
Binomials With Subtraction

So far, we've been using the formula for the square of a binomial involving addition: (a + b)^2 = a^2 + 2ab + b^2 There is also a similar formula that we can use to compute the square of a binomial involving subtraction. The formula is (a - b)^2 = a^2 - 2ab + b^2.

For example, to compute the square (x-3)^2, we can substitute a=x and b=3 into the formula and simplify:

\begin{align} (x-3)^2 & = \\[5pt] (x)^2 - 2 (x) (3) + (3)^2 & = \\[5pt] x^2 -6 x + 9 & \end{align}

We can check that the formula gives the correct result by working out the multiplication manually:

\eqalign{ (x - 3)^2 &= \\ (x - 3)(x - 3) &= \\ x(x - 3) - 3(x - 3) &= \\ x^2 - 3x - 3x + 9 &= \\ x^2 - 6x + 9 &\, {\color{green}{\checkmark}} }

Again, we get the same result.

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Example: Squaring Binomials With Subtraction

Expand (3p - 5)^2.

EXPLANATION

We will use the following formula:

(a-b)^2 = a^2 - 2ab + b^2

Substituting a=3p and b=5, we have \eqalign{ (3p - 5)^2 &= \\ (3p)^2 - 2(3p)(5) + 5^2 &= \\ 9p^2 - 30p + 25. }

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Practice: Squaring Binomials With Subtraction

$(y - 7)^2 =$

a
 $y^2-14$
b
 $y^2-49$
c
 $2(y-7)$
d
 $y^2-14y+49$
e
 $y^2-7^2$
Practice: Squaring Binomials With Subtraction

$(5p - 2q)^2=$

a
b
c
d
e
Practice: Squaring Binomials With Subtraction

$\left(-x - 3\right)^2 \,=\,$

a
b
c
d
e
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