# How do you multiply (x + -1y)(x + y)?

Sep 1, 2016

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

#### Explanation:

$\textcolor{b l u e}{\text{Using the shortcut with a bit of explanation}}$

It is the case that $1 y$ is written as just $y$

We also have $+ -$. When two signs are next to each other and they are different the result is -. So $+ - 1 y \to - y$ giving

$\left(x + - 1 y\right) \left(x + y\right) \text{ "->" } \left(x - y\right) \left(x + y\right)$

The shortcut is to know that if you have the form ${a}^{2} - {b}^{2}$ then this works out to be the same as $\left(a - b\right) \left(a + b\right)$

That is the condition in this question so $\left(x - y\right) \left(x + y\right) = {x}^{2} - {y}^{2}$

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$\textcolor{b l u e}{\text{Demonstration that this is true}}$

Consider:$\text{ } \textcolor{b l u e}{\left(x - y\right)} \textcolor{b r o w n}{\left(x + y\right)}$

Multiply everything inside the right hand side bracket by everything inside the left hand side bracket.

color(brown)(color(blue)(x)(x+y) color(blue)(-y)(x+y)

Notice that the minus follow the $\textcolor{b l u e}{y}$ in $\textcolor{b l u e}{- y}$

So we have:
color(brown)(color(blue)(x)(x+y)" " color(blue)(-y)(x+y)
${x}^{2} + x y \text{ } - x y - {y}^{2}$

${x}^{2} + 0 - {y}^{2}$

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

Sep 1, 2016

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

#### Explanation:

Each term in the 2nd bracket must be multiplied by each term in the 1st bracket.

That is $\left(\textcolor{red}{x - y}\right) \left(x + y\right) = \textcolor{red}{x} \left(x + y\right) \textcolor{red}{- y} \left(x + y\right)$

now distribute the brackets

$= {x}^{2} + \cancel{x y} - \cancel{x y} - {y}^{2} = {x}^{2} - {y}^{2}$