# The area of a rectangle is 20 cm^2. The length is 8 more than the width. How do you find the width in cm?

Jun 11, 2017

The width of the rectangle is $2$ cm.

#### Explanation:

Let's first recall the formula for the area of a rectangle:

$A = l w$

We can first substitute $20$ into the area part of the equation since we already know its value:

$20 = l w$

Now what? We have two variables but how do we find either of them? Looking back at the question, we see that "the length is 8 more than the width." Voila! Now we can write the length in terms of the width! So now we can switch out $l$ for $w + 8$:

$20 = \left(w + 8\right) w$

Now, using the Distributive Property, we can expand out the right side of the equation:

$20 = {w}^{2} + 8 w$

We can subtract $20$ from both sides to obtain a classic quadratic equation:

${w}^{2} + 8 w - 20 = 0$

Now, we can use factoring to simplify this problem. So our task now is to find two factors of $- 20$ such that they add up to $8$. We will list out all the factors of $- 20$ and then mark out the pair that adds up to $8$ (I will use red for the pairs that don't work and blue for the pair that does work):

$\textcolor{red}{- 1 \mathmr{and} 20}$

$\textcolor{red}{1 \mathmr{and} - 20}$

$\textcolor{red}{- 4 \mathmr{and} 5}$

$\textcolor{red}{4 \mathmr{and} - 5}$

$\textcolor{b l u e}{- 2 \mathmr{and} 10}$

$\textcolor{b l u e}{2 \mathmr{and} - 10}$

We found our pair! 2 and -10 or -2 and 10! So now, we can write the quadratic equation in an $\left(w + a\right) \left(w + b\right)$ format and make it equal to $0$:

$\left(w - 2\right) \left(w + 10\right) = 0$

And since the product of $\left(w - 2\right)$ and $\left(w + 10\right)$ is 0, one of them must be equal to $0$. Let's start by assuming that $\left(w - 2\right)$ is equal to $0$ and solve for $w$:

$\left(w - 2\right) = 0$
$w = 2$

Now let's assume that $\left(w + 10\right)$ is equal to $0$ and solve for $w$:

$\left(w + 10\right) = 0$
$w = - 10$

Now, we can see that $w$ could be either $- 10$ or $2$. However, there is no such thing as negative length so the possibility that $w = - 10$ is disqualified and $w = 2$ is the answer.