Question

the sum of squares of two consecutive natural numbers is 545.find the numbers.​

Answer 1

let us denote the two consecutive integers as "x" and "x + 1"

sum of their squares = 545

therefore (x)^2 + (x + 1)^2 = 545

x^2 + x ^2 + 1 + 2x = 545

2x^2 + 2x + 1 = 545

2x^2 + 2x - 544 = 0

2(x^2 + x - 272) = 0

=> x^2 + 17x - 16x - 272 = 0

=> x(x + 17) - 16(x + 17) = 0

=> (x + 17) (x - 16) = 0

x = -17 or x = 16

if x = -17 then it's consecutive no. is -16.

if x = 16 then it's consecutive integer is 17.

Answer 2

Let the first integer be x and the other be x+1

it is given that the sum of their squares is 545

x²+x²+2x+1 = 545

2x²+2x-544 = 0

x²+x-272 = 0

by Sridharacharya formula,

{-1 ± √(1+1088)}/2 = -17 or 16

therefore the answers are -17 or 16