A rectangle has a length of \( x \) units and a width of \( (x-15) \) units. If the rectangle has an area of 76 square units, what is the value of \( x \) ?

To find the value of \( x \), we can use the formula for the area of a rectangle: \[ \text{Area} = \text{length} \times \text{width} \] Given that the length is \( x \) and the width is \( (x-15) \), we can set up the equation: \[ x(x - 15) = 76 \] Expanding this, we have: \[ x^2 - 15x = 76 \] Now, we can rearrange this into standard quadratic form: \[ x^2 - 15x - 76 = 0 \] Next, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -15 \), and \( c = -76 \). Calculating the discriminant: \[ b^2 - 4ac = (-15)^2 - 4(1)(-76) = 225 + 304 = 529 \] Now, applying the quadratic formula: \[ x = \frac{-(-15) \pm \sqrt{529}}{2(1)} = \frac{15 \pm 23}{2} \] Calculating the two possible values for \( x \): 1. \( x = \frac{15 + 23}{2} = \frac{38}{2} = 19 \) 2. \( x = \frac{15 - 23}{2} = \frac{-8}{2} = -4 \) (not a valid length) Thus, the value of \( x \) is: \[ \boxed{19} \]