Question

a rectangle has a length that is 5 inches greater than its width, and its area is 104 square inches. the equation (x + 5)x = 104 represents the situation, where x represents the width of the rectangle.
(x + 5)x = 104
x² + 5x - 104 = 0
determine the solutions of the equation. what solution makes sense for the situation?
x =
what are the dimensions of the rectangle?
width = inches
length = inches

Explanation:

Step1: Solve the quadratic equation \(x^2 + 5x - 104 = 0\)

We can factor the quadratic equation. We need two numbers that multiply to \(-104\) and add up to \(5\). The numbers are \(13\) and \(-8\) because \(13\times(-8)= -104\) and \(13 + (-8)=5\). So the equation factors as \((x + 13)(x - 8)=0\).

Step2: Find the solutions for \(x\)

Setting each factor equal to zero gives \(x + 13 = 0\) or \(x - 8 = 0\). Solving these, we get \(x=-13\) or \(x = 8\). Since the width cannot be negative, we discard \(x=-13\). So \(x = 8\).

Step3: Find the length

The length is \(x + 5\). Substituting \(x = 8\), we get length \(= 8 + 5=13\).

Answer:

For the equation solution, \(x=\boldsymbol{8}\).
For the dimensions:
width \(=\boldsymbol{8}\) inches
length \(=\boldsymbol{13}\) inches