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?

Explanation:

Step1: Define length and area formula

Let width be \( w \), then length \( l = 2w + 1 \). The area of a rectangle is \( A = l \times w \), so substitute \( l \) and \( A = 21 \):
\( w(2w + 1) = 21 \)

Step2: Expand and form quadratic equation

Expand the left side: \( 2w^2 + w = 21 \).
Rearrange to standard quadratic form: \( 2w^2 + w - 21 = 0 \)

Step3: Solve the quadratic equation

Factor the quadratic (or use quadratic formula). Factoring:
\( 2w^2 + 7w - 6w - 21 = 0 \)
\( w(2w + 7) - 3(2w + 7) = 0 \)
\( (w - 3)(2w + 7) = 0 \)

Solutions: \( w = 3 \) (since width can’t be negative, \( 2w + 7 = 0 \) gives \( w = -\frac{7}{2} \), invalid).

Step4: Find length

Substitute \( w = 3 \) into \( l = 2w + 1 \):
\( l = 2(3) + 1 = 7 \)

Answer:

The length is 7 inches.