Question

The rectangle below has an area of
\[x^2+13x+36\] square meters and a length of
\[x+9\] meters.
What expression represents the width of the rectangle?

Answer 1

The area of a rectangle is calculated using the formula:

\[
\text{Area} = \text{Length} \times \text{Width}
\]

In this case, the area is given as \( x^2 + 13x + 36 \) square meters, and the length is given as \( x + 9 \) meters. To find the width, we can rearrange the area formula to solve for the width:

\[
\text{Width} = \frac{\text{Area}}{\text{Length}}
\]

Substituting in the given expressions for area and length, we have:

\[
\text{Width} = \frac{x^2 + 13x + 36}{x + 9}
\]

Next, we need to simplify the expression. We can factor the numerator \( x^2 + 13x + 36 \).

To factor \( x^2 + 13x + 36 \), we look for two numbers that multiply to \( 36 \) (the constant term) and add up to \( 13 \) (the coefficient of \( x \)). The numbers \( 9 \) and \( 4 \) meet these criteria. Thus, we can factor the quadratic as follows:

\[
x^2 + 13x + 36 = (x + 9)(x + 4)
\]

Now we can substitute this factorization back into the width expression:

\[
\text{Width} = \frac{(x + 9)(x + 4)}{x + 9}
\]

Assuming \( x + 9 \neq 0 \), we can cancel \( x + 9 \) from the numerator and denominator:

\[
\text{Width} = x + 4
\]

Thus, the expression that represents the width of the rectangle is:

\[
\boxed{x + 4}
\]