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

Hint: Here, we need to find out two numbers provided the difference of squares of two numbers is $180$ and the square of the smaller number is $8$ times the larger number. We take the square of the smaller number as ${{x}^{2}}$ and that of the larger number as ${{y}^{2}}$ . We get two equations,
${{y}^{2}}-{{x}^{2}}=180....\left( i \right)$
${{x}^{2}}=8y....\left( ii \right)$
Solving them, we get the answers.

Complete step-by-step answer:
The entire problem is about the squares of two numbers. So, let the square of the first number (which is the smaller number) be ${{x}^{2}}$ and the square of the second number (which is the larger number) be ${{y}^{2}}$ . In the problem, it is given that the difference of squares of two numbers is $180$ . This means that,
${{y}^{2}}-{{x}^{2}}=180....\left( i \right)$
Also, it is said in the same problem that the square of the smaller number is $8$ times the larger number. This means that,
${{x}^{2}}=8y....\left( ii \right)$
Let us now find out the value of ${{y}^{2}}$ from the second equation. Doing so, we get,
$\begin{align}
  & \Rightarrow y=\dfrac{{{x}^{2}}}{8} \\
 & \Rightarrow {{y}^{2}}=\dfrac{{{x}^{4}}}{64} \\
\end{align}$
We now substitute this value of ${{y}^{2}}$ in the first equation to get,
$\begin{align}
  & \Rightarrow \dfrac{{{x}^{4}}}{64}-{{x}^{2}}=180 \\
 & \Rightarrow {{x}^{4}}-64{{x}^{2}}=11520 \\
 & \Rightarrow {{x}^{4}}-64{{x}^{2}}-11520=0 \\
\end{align}$
Applying Sridhar Acharya formula, we get the values as,
$\begin{align}
  & {{x}^{2}}=\dfrac{-\left( -64 \right)\pm \sqrt{{{\left( -64 \right)}^{2}}-4\left( -11520 \right)}}{2} \\
 & \Rightarrow {{x}^{2}}=\dfrac{64\pm 224}{2} \\
 & \Rightarrow {{x}^{2}}=144,-80 \\
\end{align}$
Now, since ${{x}^{2}}$ is a square, its value must be positive. So, ${{x}^{2}}=144$ . This gives, from equation (i), ${{y}^{2}}=180+144=324$ . The numbers will be,
$\begin{align}
  & x=\sqrt{144}=12 \\
 & y=\sqrt{324}=18 \\
\end{align}$
Thus, we can conclude that the numbers are $12,18$ .

Note: There are many mistakes that we can make here. At first, we may ignore the squares and write the equations in terms of the numbers. Secondly, we may write the squares of the numbers as our final answer. All of these lead to a wrong answer.