# A rectangle has a perimeter if 46 cm and an area of 120 cm^2. How do you find its dimensions by writing an equation and using the quadratic formula to solve it?

Apr 7, 2018

The dimensions are 8 cm and 15 cm.

#### Explanation:

Let the length be $x$ and the width be $y$

The perimeter of the rectangle is $2 x + 2 y$

The area of the rectangle is $x y$

We now have two equations,

$2 x + 2 y = 46 \mathmr{and} x + y = 23$ we’ll call this equation (1)

And $x y = 120$ equation (2)

From (2), $y = \frac{120}{x}$, substitute this into (1)

$\therefore x + \frac{120}{x} = 23$

${x}^{2} + 120 = 23 x$ multiply both sides by $x$

${x}^{2} - 23 x + 120 = 0$ subtract $23 x$ from both sides

$\left(x - 8\right) \left(x - 15\right) = 0$ factorise

$x = 8 \mathmr{and} 15$ solve linear equations

From (2), when $x = 8 , 8 y = 120 \implies y = 15$

When $x = 15 , 15 y = 120 \implies y = 8$

$\therefore$ the dimensions are $8$ cm and $15$ cm.