Question

find two. consecutive natural numbers the sum of whose square is 145​

Answer 1

Let one number be x

Then other number = x+1

It is given that the sum of square of those numbers is 145.

Therefore,  

x² + (x+1)² = 145.

x² + x² + 2x + 1 = 145              {(a+b)²= a² + 2ab + b²}

2x² + 2x - 144 = 0  

Dividing both sides by 2, we obtain

x² + x - 72 = 0

x² - 8x + 9x - 72 = 0

x(x-8) + 9 (x-8) = 0

(x+9) (x-8) = 0

x = -9  or x=8  

As x = -9 can't be answer because it is not a natural number.

Hence, x = 8.

And, x+1 = 9.

Justification:-

                        8² + 9²

                      = 64 + 81 = 145

Hence, it is proved.

Hope it helps