# How do you multiply (x-2y)^2?

Mar 3, 2018

See a solution process below:

#### Explanation:

This is a special form of the quadratic:

${\left(\textcolor{red}{a} - \textcolor{b l u e}{b}\right)}^{2} = \left(\textcolor{red}{a} - \textcolor{b l u e}{b}\right) \left(\textcolor{red}{a} - \textcolor{b l u e}{b}\right) = {\textcolor{red}{a}}^{2} - 2 \textcolor{red}{x} \textcolor{b l u e}{b} + {\textcolor{b l u e}{b}}^{2}$

Let:

• $\textcolor{red}{a} = x$

• $\textcolor{b l u e}{b} = 2 y$

Substituting gives:

${\left(\textcolor{red}{x} - \textcolor{b l u e}{2 y}\right)}^{2} \implies$

$\left(\textcolor{red}{x} - \textcolor{b l u e}{2 y}\right) \left(\textcolor{red}{x} - \textcolor{b l u e}{2 y}\right) \implies$

${\textcolor{red}{x}}^{2} - \left(2 \times \textcolor{red}{x} \times \textcolor{b l u e}{2 y}\right) + {\textcolor{b l u e}{\left(2 y\right)}}^{2} \implies$

${\textcolor{red}{x}}^{2} - 4 \textcolor{red}{x} \textcolor{b l u e}{y} + {\textcolor{b l u e}{4 y}}^{2}$

Mar 3, 2018

${x}^{2} - 4 x y + 4 {y}^{2}$

#### Explanation:

We can rewrite ${\left(x - 2 y\right)}^{2}$ as $\textcolor{b l u e}{\left(x - 2 y\right) \left(x - 2 y\right)}$. The expression I have in blue, we can use FOIL to multiply.

$\left(x - 2 y\right) \left(x - 2 y\right)$

FOIL tells us that we multiply the first terms, outside terms, inside terms and last terms respectively. We get:

• Firsts $\left(x \cdot x\right) = {x}^{2}$
• Outsides $\left(x \cdot - 2 y\right) = - 2 x y$
• Insides $\left(- 2 y \cdot x\right) = - 2 x y$
• Lasts $\left(- 2 y \cdot - 2 y\right) = 4 {y}^{2}$

NOTE:Whenever we're multiplying binomials, we can use FOIL

Our new expression is as follows:

${x}^{2} - 2 x y - 2 x y + 4 {y}^{2}$

We can combine like terms to get:

${x}^{2} - 4 x y + 4 {y}^{2}$

Alternatively, we could have separated $\left(x - 2 y\right) \left(x - 2 y\right)$ into two expressions if you're not a FOIL person. This is the same as doing:

$x \left(x - 2 y\right) - 2 y \left(x - 2 y\right)$

We can distribute the $x$ and the $- 2 y$ respectively to get:

${x}^{2} - 2 x y - 2 x y + 4 {y}^{2}$

And we could combine like terms to get:

${x}^{2} - 4 x y + 4 {y}^{2}$