Question

the area of a rectangle is $21\\ \text{in}^2$. the length is 2 times the width plus 1. use $w$ for width. what is the length in inches? equation: $2w^2 + w - 21 = 0$ solution: length =? inches

Explanation:

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

We can use the quadratic formula \(w=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\) for a quadratic equation \(ax^2+bx + c = 0\). Here, \(a = 2\), \(b = 1\), \(c=- 21\).
First, calculate the discriminant \(\Delta=b^2 - 4ac=(1)^2-4\times2\times(-21)=1 + 168 = 169\)
Then, \(w=\frac{-1\pm\sqrt{169}}{2\times2}=\frac{-1\pm13}{4}\)
We have two solutions for \(w\):
\(w_1=\frac{-1 + 13}{4}=\frac{12}{4}=3\) and \(w_2=\frac{-1-13}{4}=\frac{-14}{4}=-3.5\)
Since the width cannot be negative, we take \(w = 3\) inches.

Step2: Find the length

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

Answer:

7