Question

Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the squares of the other, two by 60. Find the numbers.

Answer 1

Hint: For solving this problem, first let the three consecutive natural numbers as x - 1, x, x + 1. Now, by using the fact that the square of x exceeds the difference of the square of the other two numbers by 60, we formulate an equation in x. By solving this equation, we get the value of x.

Complete step-by-step solution -

According to the problem statement, we are given three consecutive natural numbers. So, we let the three consecutive natural numbers as x - 1, x, x + 1.
First, the square of all the numbers can be expressed as \[{{\left( x-1 \right)}^{2}},\text{ }{{x}^{2}},\text{ }{{\left( x+1 \right)}^{2}}\].
Some helpful identity evaluating the square are
$\begin{align}
  & {{(a+b)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab \\
 & {{(a-b)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab \\
\end{align}$
Now as stated in question, the square of the middle number exceeds the difference of the square of other two by 60, we formulate the desired equation as
$\begin{align}
  & {{x}^{2}}-\left[ {{\left( x+1 \right)}^{2}}-{{\left( x-1 \right)}^{2}} \right]=60 \\
 & {{x}^{2}}-\left[ {{x}^{2}}+1+2x-{{x}^{2}}-1+2x \right]=60 \\
 & {{x}^{2}}-\left( 4x \right)=60 \\
 & {{x}^{2}}-4x-60=0 \\
\end{align}$
Now, by using splitting the middle term method to factorise the above equation, we get
$\begin{align}
  & {{x}^{2}}-10x+6x-60=0 \\
 & x\left( x-10 \right)+6\left( x-10 \right)=0 \\
 & \left( x+6 \right)\left( x-10 \right)=0 \\
 & x=-6,10 \\
\end{align}$
Therefore, the obtained values of x are -6 and 10.
Since natural numbers include only positive numbers, so neglecting the negative value of x.
Hence, the final numbers are: 9, 10, 11.

Note: The key step for solving this problem is the way of expressing three consecutive natural numbers. Students must have full concept knowledge of natural numbers so that they can neglect the final negative value. If we consider the negative value then two possible sets are obtained which is an incorrect answer.