Question

The difference of squares of two numbers is $180$. The square of the smaller number is $8$ times the larger number. Find the two numbers.

Answer 1

Step 1. Determine the required equations.

Assume that the numbers are $x\mathrm{and}y$.

The equation to represent the difference of their squares is:

$x^2-y^2=180$

The equation to represent the second relation is:

$y^2=8x$

Step 2. Substitute and solve.

Substitute $y^2=8x$ into $x^2-y^2=180$ and solve for $y$:

$\begin{array}{rcl}{x}^2-8x& =& 180\\ & \Rightarrow & x^2-8x-180=0\\ & \Rightarrow & \left(x-18\right)\left(x+10\right)=0\end{array}$

$\begin{array}{rcl}& \Rightarrow & x=18\mathrm{or}\mathrm{x}=-10\end{array}$

Assume that $x=-10$.

Substitute $x=-10$ into $y^2=8x$ and solve:

$\begin{array}{rcl}{y}^2& =& 8\times \left(-10\right)\\ & \Rightarrow & y^2=-80\end{array}$

The Square of a number cannot be negative, so $x=18$.

Substitute $x=18$ into $y^2=8x$ and solve:

$$\begin{gathered} \begin{array}{rcl}{y}^2& =& 8\times 18\\ & \Rightarrow & y^2=144\end{array} \\ \Rightarrow y=\pm 12 \end{gathered}$$

So, the larger number is $18$ and the smaller number is $-12\mathrm{or}12$.