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

Setting each factor equal to zero gives:
$x + 13 = 0$ or $x - 8 = 0$
Solving $x + 13 = 0$ gives $x = -13$. Solving $x - 8 = 0$ gives $x = 8$. Since the width cannot be negative, we discard $x = -13$. So, $x = 8$.

Step3: Find the length

The length is $x + 5$. Substituting $x = 8$ into this expression, we get length $= 8 + 5 = 13$.

Answer:

For the equation solution, $x = 8$.
For the dimensions:
width $= 8$ inches
length $= 13$ inches