## Elementary Algebra

Let the unknown number be $x$. This means that one-half of the number is $\frac{x}{2}$. As a result, the square of the unknown number is $x^{2}$ whereas the square of one-half of the number is $(\frac{x}{2})^{2}$. Since the difference between the square of a number and the square of one-half of the number is 243, we write the following equation and solve it: $x^{2}-(\frac{x}{2})^{2}=243$ $x^{2}-(\frac{x^{2}}{4})=243$ $\frac{4x^{2}-x^{2}}{4}=243$ $\frac{3x^{2}}{4}=243$ $3x^{2}=972$ $x^{2}=\frac{972}{3}$ $x^{2}=324$ $x=\pm\sqrt {324}$ $x=\pm 18$ Check when x=18: $18^{2}-(\frac{18}{2})^{2}=243$ $18^{2}-(9)^{2}=243$ $324-81=243$ $243=243$ The problem asks for the square of a positive number, meaning that we should disregard the negative answer, giving us 18 as our answer.