Let the width of the rectangle be \( x \). Then the length can be expressed as \( 2x - 7 \). The area of the rectangle is given by the formula:
\[ \text{Area} = \text{length} \times \text{width} \]
According to the problem, the area is 15 square meters, so we can write the equation:
\[ x(2x - 7) = 15 \]
Expanding this gives:
\[ 2x^2 - 7x = 15 \]
Rearranging the equation to set it to zero:
\[ 2x^2 - 7x - 15 = 0 \]
Now, we can solve this quadratic equation using the quadratic formula, where \( a = 2 \), \( b = -7 \), and \( c = -15 \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Calculating the discriminant:
\[ b^2 - 4ac = (-7)^2 - 4 \cdot 2 \cdot (-15) = 49 + 120 = 169 \]
Now substituting the values into the formula:
\[ x = \frac{7 \pm \sqrt{169}}{2 \cdot 2} = \frac{7 \pm 13}{4} \]
Calculating the two possible solutions:
\[ x = \frac{20}{4} = 5 \quad \text{and} \quad x = \frac{-6}{4} = -1.5 \]
Since width cannot be negative, we have:
\[ x = 5 \]
Thus, the value of \( x \) is:
\[ \boxed{5} \]
1 answer