7 and 8
Step-by-step explanation:
Consecutive numbers follow each other without a gap.
Let the consecutive intergers be:
x, x+1
The sum of their squares = 113
x² + (x + 1)² = 113
Expanding:
x² + (x + 1)(x + 1) = 113
x² + x² + x + x + 1 = 113
x² + x² + 2x + 1 = 113
2x²+ 2x + 1 = 113
subtract 1 from both sides:
2x² + 2x = 113 - 1
2x² + 2x = 112
divide both sides by 2:
2(x² + x) = 2(56)
x² + x = 56
x² + x - 56 = 0
Factorize the equation:
factorsof -56 whose sum gives - 1 are +8 and -7
x² + 8x - 7x - 56 = 0
x(x + 8) -7(x + 8) = 0
(x - 7)(x + 8) = 0
x - 7 = 0
x = 7
OR
x + 8 = 0
x = -8
Note: We are told the consecutive intergers are positive. This rules -8 as an answer as it is a negative interger.
when x , first consecutive number = 7
second consecutive number = x +1 = 7 + 1 = 8
Hence, the consecutive numbers are 7 and 8