Question

the area of a rectangle is $16\\ \text{in}^2$. the length is 3 times the width plus 2. use $w$ for width. what is the width in inches? equation: $3w^2 + 2w - 16 = 0$ solution:?

Explanation:

Step1: Identify the quadratic formula

For a quadratic equation \( ax^{2}+bx + c = 0 \), the solutions are given by \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \). Here, \( a = 3 \), \( b = 2 \), \( c=- 16 \).

Step2: Calculate the discriminant

The discriminant \( D=b^{2}-4ac \). Substitute the values: \( D=(2)^{2}-4\times3\times(-16)=4 + 192 = 196 \).

Step3: Find the square root of the discriminant

\( \sqrt{D}=\sqrt{196} = 14 \).

Step4: Apply the quadratic formula

\( w=\frac{-b\pm\sqrt{D}}{2a}=\frac{-2\pm14}{2\times3} \). We have two solutions:

• \( w_1=\frac{-2 + 14}{6}=\frac{12}{6}=2 \)
• \( w_2=\frac{-2-14}{6}=\frac{-16}{6}=-\frac{8}{3} \)

Since width cannot be negative, we take \( w = 2 \).

Answer:

The width is \( 2 \) inches.