Question

A sidewalk of equal width is being built around a square yard. What is the width of the sidewalk?

Short answer

The width of the sidewalk is:

$\mathit{x}\mathbf{=}\mathbf{36}$

Step-by-step solution

Step 1. State the concept for quadratic equation.

A quadratic equation is a second-order equation written as $a{x}^2+bx+c=0$ where a, b, and c are coefficients of real numbers and $a\ne 0$.

A quadratic equation generally has two roots, which can be equal to each other. Some quadratic expressions can be factored, which means that the equation can be written as:

$a\left(x-x_1\right)\left(x-x_2\right)=0$

where $x_1$and $x_2$are the two roots of the equation.

Step 2. Form the equation.

Here the total area is:

$A=900f{t}^2$

Again, the length of each side of the inner square is:

$d=6ft$

Assume the width of the sidewalk be x.

Hence, the area of the whole yard is given by:

${\left(x-d\right)}^2=900$

Step 3. Solve the equation.

Substitute the value and solve the equation.

$\begin{array}{c}{\left(x-d\right)}^2=900\\ {\left(x-6\right)}^2=900\\ {\left(x-6\right)}^2-900=0\\ {\left(x-6\right)}^2-{30}^2=0\\ \left(x-6-30\right)\left(x-6+30\right)=0\\ \left(x-36\right)\left(x+24\right)=0\\ x-36=0\text{or}x+24=0\\ x=36\text{or}x=-24\\ x=36,-24\end{array}$

Since the distance cannot be negative, therefore, the width of the sidewalk will be$36ft$.