Question

find two consecutive odd natural numbers the sum of whose square is 394

Answer 1

Let the two consecutive odd numbers be x and x+2.
ATQ,

X2 + (X+2)2 = 394
X2 + X2 + 4X + 22 = 394
2X2 + 4X + 4 = 394 .......( dividing the equation by 2)
X2 +2X =(394 - 4)/2
X2 + 2X = 195
X2 +2X - 195 = 0
X2 + 15X - 13X -195 = 0 .............( by factorization meathod )
X( X + 15 ) - 13( X + 15 ) = 0
( X + 15 ) ( X - 13 )
X = -15 OR X = 13

CASE 1
If X = -15, then,
X + 2 = - 13

CASE 2
If X = 13, then,
X + 2 = 15