# The length of a rectangle is 3 cm more than the width. The area is 70cm^2. How do you find the dimensions of the rectangle?

Oct 2, 2015

If we write $w$ for the width in $\text{cm}$, then $w \left(w + 3\right) = 70$.

Hence we find $w = 7$ (discarding negative solution $w = - 10$).

So width $= 7 \text{cm}$ and length $= 10 \text{cm}$

#### Explanation:

Let $w$ stand for the width in $\text{cm}$.

Then the length in $\text{cm}$ is $w + 3$ and the area in ${\text{cm}}^{2}$ is $w \left(w + 3\right)$

So:

$70 = w \left(w + 3\right) = {w}^{2} + 3 w$

Subtract $70$ from both ends to get:

${w}^{2} + 3 w - 70 = 0$

There are a variety of ways to solve this, including the quadratic formula, but we can instead recognise that we're looking for a pair of factors of $70$ which differ by $3$.

It should not take long to find $70 = 7 \times 10$ fits the bill, hence we find:

${w}^{2} + 3 w - 70 = \left(w - 7\right) \left(w + 10\right)$

So ${w}^{2} + 3 w - 70 = 0$ has two solutions, viz $w = 7$ and $w = - 10$.

Since we're talking about lengths, we can ignore the negative solution leaving $w = 7$. That is the width is $7 \text{cm}$ and the length is $10 \text{cm}$.