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, \(a = 3\), \(b = 2\), and \(c=- 16\).

Step2: Calculate the discriminant

The discriminant \(D=b^{2}-4ac\). Substitute \(a = 3\), \(b = 2\), \(c=-16\) into it:
\(D=(2)^{2}-4\times3\times(-16)=4 + 192=196\)

Step3: Find the square root of the discriminant

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

Step4: Apply the quadratic formula to find \(w\)

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

• For the plus sign: \(w=\frac{-2 + 14}{6}=\frac{12}{6}=2\)
• For the minus sign: \(w=\frac{-2-14}{6}=\frac{-16}{6}=-\frac{8}{3}\)

Since width cannot be negative, we discard \(w =-\frac{8}{3}\).

Answer:

The width is \(2\) inches.