Question

If the area of the rectangle below is 39 square feet find the value of x
Length x+8 and width x-2

Answer 1

To find the value of \( x \) for the rectangle with an area of 39 square feet, we can use the formula for the area of a rectangle:

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

In this case, the length is \( x + 8 \) and the width is \( x - 2 \). Therefore, we can set up the equation:

\[
(x + 8)(x - 2) = 39
\]

Now, we can expand the left-hand side:

\[
x^2 - 2x + 8x - 16 = 39
\]

This simplifies to:

\[
x^2 + 6x - 16 = 39
\]

Next, we move 39 to the left-hand side to set the equation to zero:

\[
x^2 + 6x - 16 - 39 = 0
\]

Simplifying this gives:

\[
x^2 + 6x - 55 = 0
\]

Now, we need to factor this quadratic equation. We are looking for two numbers that multiply to \(-55\) and add to \(6\). The numbers \(11\) and \(-5\) satisfy this requirement:

\[
(x + 11)(x - 5) = 0
\]

Setting each factor equal to zero gives us the possible solutions for \( x \):

1. \( x + 11 = 0 \) ⟹ \( x = -11 \)
2. \( x - 5 = 0 \) ⟹ \( x = 5 \)

Since \( x \) represents a length, it must be positive. Thus, we discard \( x = -11 \) and conclude that:

\[
x = 5
\]

To verify, we can calculate the length and width with \( x = 5 \):

- Length: \( 5 + 8 = 13 \) feet
- Width: \( 5 - 2 = 3 \) feet

Calculating the area:

\[
\text{Area} = 13 \times 3 = 39 \text{ square feet}
\]

This matches the given area, confirming that the solution is correct.

The value of \( x \) is \( \boxed{5} \).