Using Factoring to Solve Problems

Example

Consider the rectangle in Example C with sides of length \(x + 5\) and \(x - 3\). What is \(x\) if the area of the rectangle is now 20?

Make a sketch of this situation:

Using the formula Area = length \(\times\) width, we have \((x+5)(x-3)=20\).

In order to solve, we must write the polynomial in standard form. Distribute, collect like terms and rewrite:

\(x^2+2x-15 =20\\ x^2+2x-35 =0\)

Factor by finding two numbers that multiply to -35 and add to 2. List some numbers that multiply to -35:

\( -35 = -7 \cdot 5 \quad\quad \text{and} \quad -7 + 5 = -2 \\ -35 = 7 \cdot (-5) \quad \text{and} \quad 7 + (-5) = 2 \)

The expression factors as \((x+7)(x-5)=0\).

Set each term equal to zero and solve:

\( x+7=0 \quad \quad x-5=0\\ \quad \quad \quad \quad \text{or}\\ \underline{\underline{x=-7}} \quad \quad\underline{\underline{x=5}} \)

Since we are looking for positive numbers the answer must be \(x = 5\). So the width is \(x - 3 = 2\) and the length is \(x + 5 = 10\).

Check: \(2 \cdot 10 = 20\), so the answer checks.