91Ó°ÊÓ

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 $a≠0$.

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.

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:

${\left(x−d\right)}^2=900$

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$36ft$.