Question

find the two consecutive numbers whose squares have the sum 85

Answer 1

let one number be x
therefore other number = x+1
according to question = x^2 + (x+1)^2 = 85
                                   = x^2 + x^2 +1 + 2x = 85
                                   = 2x^2 + 2x + 1 = 85
                                   = 2x^2 + 2x - 84 = 0
                                   = x^2 + x - 42 = 0
                                   = x^2 + 7x - 6x - 42 = 0
                                   = x(x+7) - 6(x+7) = 0
                                   = (x-6)(x+7) = 0
therefore x = 6 , -7
if x = 6
then the numbers are 6 and 7
if x = -7
then the numbers are -7 and -6