Exercise 9 Page 576

When we are given the area of a rectangle, we generally need to use the formula for the area of a rectangle. In this case we need to write expressions for the length, $\ell ,$ and width, $w.$

Verbal Expression	Algebraic Expression
the length is $10\text{ ft}$ more than the width	$\ell =10+w$

Now that we have an algebraic expression, we can substitute. Then, let's see where we can go from there. Now, we have a quadratic equation that is equal to zero. Let's look for two factors of $\text{-}144$ that have a sum of $10.$ Since $ac$ is negative, the sign of the factors will be different. Furthermore, since $b$ is positive, the factor with the larger magnitude will be positive too.

Factors	Sum of factors
$\text{-}1,144$	$143$
$\text{-}2,72$	$70$
$\text{-}3,48$	$45$
$\text{-}4,36$	$32$
$\text{-}6,24$	$18$
$\text{-}8,18$	$10$

We have found our factors that have a sum of $10.$ Since our leading coefficient is $1,$ we can use $\text{-}8$ and $18$ directly as part of the factoring. Let's set each factor equal to zero and solve for $w.$ Since $w$ represents a measurement of distance, we can ignore the negative solution and say the width of the rectangle is $8\text{ ft}.$ To find the length, we can add $10$ to the width and get $18\text{ ft}.$