Question

find the value of x. show all work. 22 area of the rectangle is 30 sq. ft.

Explanation:

Step1: Recall area formula

The area formula for a rectangle is $A = length\times width$. Here, $A = 30$, length $=3x + 1$, and width $=x$. So we have the equation $x(3x + 1)=30$.

Step2: Expand the equation

Expand $x(3x + 1)$ to get $3x^{2}+x=30$. Then rewrite it in standard quadratic - form $3x^{2}+x - 30=0$.

Step3: Solve the quadratic equation

For a quadratic equation $ax^{2}+bx + c = 0$ (here $a = 3$, $b = 1$, $c=-30$), we can use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. First, calculate the discriminant $\Delta=b^{2}-4ac=(1)^{2}-4\times3\times(-30)=1 + 360 = 361$. Then $x=\frac{-1\pm\sqrt{361}}{2\times3}=\frac{-1\pm19}{6}$.

Step4: Find the values of x

We have two solutions:
$x_1=\frac{-1 + 19}{6}=\frac{18}{6}=3$ and $x_2=\frac{-1-19}{6}=\frac{-20}{6}=-\frac{10}{3}$. But since the width of a rectangle cannot be negative, we discard $x =-\frac{10}{3}$.

Answer:

$x = 3$