Question

the area of a rectangle is 21 in². the length is 2 times the width plus 1. use w for width. what is the length in inches? equation: 2w² + w - 21 = 0 solution: length =? inches

Explanation:

Step1: Solve the quadratic equation \(2w^2 + w - 21 = 0\)

We can factor the quadratic equation. We need two numbers that multiply to \(2\times(-21)= -42\) and add up to \(1\). The numbers are \(7\) and \(-6\).
So, we rewrite the middle term:
\(2w^2 + 7w - 6w - 21 = 0\)
Group the terms:
\((2w^2 + 7w) + (-6w - 21) = 0\)
Factor out the common factors from each group:
\(w(2w + 7) - 3(2w + 7) = 0\)
Now, factor out \((2w + 7)\):
\((2w + 7)(w - 3) = 0\)
Set each factor equal to zero:
\(2w + 7 = 0\) or \(w - 3 = 0\)
Solving \(2w + 7 = 0\) gives \(w = -\frac{7}{2}\), but width cannot be negative.
Solving \(w - 3 = 0\) gives \(w = 3\).

Step2: Find the length

The length \(l\) is given by \(l = 2w + 1\). Substitute \(w = 3\) into this formula:
\(l = 2\times3 + 1 = 6 + 1 = 7\)

Answer:

\(7\)