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

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Question

Find two consecutive positive integers, the sum of whose square is $$365.$$

Video Solution
Ex 4.2 | Question 4

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 \\ [\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\end{align}

\begin{align} x - 13&=0 & x + 14 = 0\\x& = 13 & 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 $$14.$$

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