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

## Question

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 \(14.\)