Question

If the area of the rectangle below is 39 square feet, find the value of x. Just type the number
one side is x - 2 and the other is x - 8

Answer 1

To find the value of \( x \), we can use the formula for the area of a rectangle, which is given by:

\[
\text{Area} = \text{length} \times \text{width}
\]

In this case, the area is 39 square feet, one side is \( x - 2 \), and the other side is \( x - 8 \). Therefore, we can set up the equation:

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

Now, we will expand the left-hand side:

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

This simplifies to:

\[
x^2 - 10x + 16 = 39
\]

Next, we will bring all terms to one side of the equation:

\[
x^2 - 10x + 16 - 39 = 0
\]

This simplifies to:

\[
x^2 - 10x - 23 = 0
\]

Now, we will use the quadratic formula to find the values of \( x \):

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1, b = -10, c = -23 \).

Calculating the discriminant:

\[
b^2 - 4ac = (-10)^2 - 4(1)(-23) = 100 + 92 = 192
\]

Now substituting values into the quadratic formula:

\[
x = \frac{10 \pm \sqrt{192}}{2}
\]

Calculating \( \sqrt{192} \):

\[
\sqrt{192} = \sqrt{64 \times 3} = 8\sqrt{3}
\]

Now substituting back to find \( x \):

\[
x = \frac{10 \pm 8\sqrt{3}}{2}
\]

This simplifies to:

\[
x = 5 \pm 4\sqrt{3}
\]

Calculating the approximate numerical values:

1. \( x = 5 + 4\sqrt{3} \approx 5 + 6.928 \approx 11.928 \)
2. \( x = 5 - 4\sqrt{3} \approx 5 - 6.928 \approx -1.928 \)

Since \( x \) must be a positive value (both expressions \( x - 2 \) and \( x - 8 \) must be positive for the sides), we take:

\[
x \approx 11.928
\]

Thus, the value of \( x \) is approximately:

\[
\boxed{11.928}
\]

If you are looking for just the integer part, then the answer would be \( 11 \).