Ex.4.2 Q4 Quadratic Equations Solutions - NCERT Maths Class 10

Find two consecutive positive integers, the sum of whose square is \(365.\)

Text Solution

What is known?

Sum of squares of these two consecutive integers is \(365.\)

What is Unknown?

Two consecutive positive integers.

Reasoning:

Let the first integer be \(x.\)

The next consecutive positive integer will be \(x + 1.\)

\[x^2+\left( x+1 \right)^2=365\]

Steps:

\[\begin{align}{x^2} + \,{\left( {x + 1} \right)^2} &= 365\\{x^2} + \left( {{x^2} + 2x + 1} \right) &= 365 \qquad \because {\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\\2{x^2} + 2x + 1 &= 365\\
2{x^2} + 2x + 1 - 365 &= 0\\2{x^2} + 2x - 364 &= 0\\2({x^2} + x - 182)\,& = \,0\\
x + x-182 &= 0\\x + 14x-13x-182 &= 0\\x\left( {x + 14} \right)-13\left( {x + 14} \right) &= 0\\
\left( {x - 13} \right)\left( {x + 14} \right) &= 0\\x - 13=0 & \qquad x + 14 = 0\\x = 13& \qquad \qquad \;x =- 14\end{align}\]

Value of \(x\) cannot be negative (because it is given that the integers are positive).

\[\therefore \,\,~x=13 \qquad x\text{ }+\text{ }1\text{ }=\text{ }14\]

The two consecutive positive integers are \(13\) and \(14.\)