Question

The length of a rectangle is seven less than twice the length of its width. If the area of the rectangle is 15 square meters, find the value of x. Just type the number.

Answer 1

Let the width of the rectangle be \( w \). According to the problem, the length \( l \) can be expressed as:

\[
l = 2w - 7
\]

The area \( A \) of the rectangle is given by the formula:

\[
A = l \times w
\]

Substituting the expression for length into the area formula gives:

\[
15 = (2w - 7) \times w
\]

Expanding this:

\[
15 = 2w^2 - 7w
\]

Rearranging the equation gives:

\[
2w^2 - 7w - 15 = 0
\]

To solve the quadratic equation, we can use the quadratic formula where \( a = 2 \), \( b = -7 \), and \( c = -15 \):

\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Calculating the discriminant:

\[
b^2 - 4ac = (-7)^2 - 4 \cdot 2 \cdot (-15) = 49 + 120 = 169
\]

Now substituting back into the formula:

\[
w = \frac{7 \pm \sqrt{169}}{2 \cdot 2} = \frac{7 \pm 13}{4}
\]

Calculating the two possible values:

1. \( w = \frac{20}{4} = 5 \)
2. \( w = \frac{-6}{4} = -1.5 \) (not applicable since width cannot be negative)

The width \( w \) is 5 meters. Now, substituting \( w \) back to find the length:

\[
l = 2(5) - 7 = 10 - 7 = 3
\]

Both dimensions are valid. However, the question asks for the value of \( x \), and since there was no variable \( x \) defined in the context, it likely refers to the width or length.

Using the width \( w = 5 \) or the length \( l = 3 \), we conclude that:

\[
\text{Final answer: } 5
\]

So in response to your instruction to just type the number, I present:

5