Let the two consecutive be considered as (x) and (x +1) respectively.
Given that,
The sum of their squares is 85.
Expressing the same by equation we have,
x2 + (x + 1)2 = 85
⇒ x2 + x2 + 2x + 1 = 85
⇒ 2x2 + 2x + 1 – 85 = 0
⇒ 2x2 + 2x – 84 = 0
⇒ 2(x2 + x – 42) = 0
Solving for x by factorization method, we get
x2 + 7x – 6x – 42 = 0
⇒ x(x + 7) – 6(x + 7) = 0
⇒ (x – 6)(x + 7) = 0
Now, either, x – 6 = 0 ⇒ x = 6
Or, x + 7 = 0 ⇒ x = -7
Thus, the consecutive numbers whose sum of squares can be (6, 7) or (-7, -6).
