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 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: $w_1=\frac{-1 + 13}{4}=\frac{12}{4}=3$ and $w_2=\frac{-1-13}{4}=\frac{-14}{4}=-3.5$. Since width cannot be negative, we take $w = 3$.

Step2: Find the length

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

Answer:

7