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Question 116943: Number problems. Find two consecutive positive integers such that the sum of their squares is 85.
Answer by ganesh(20)   (Show Source):
You can put this solution on YOUR website! Let x and x +1 be two consecutive positive numbers.
The square of the first number is x^2.
The square of the second number is, (x+1)^2 = x^2 + 2x + 1.
The sum of the squares of the two numbers = x^2 + x^2 + 2x + 1
= 2x^2 + 2x + 1
Given that, the sum of the squares is 85.
Therefore, 2x^2 + 2x + 1 = 85.
Or, 2x^2 + 2x - 84 = 0.
Solve this quadratic equation.
2x^2 + 14x -12x - 84 = 0
2x (x +7) -12 (x +7) = 0
(x + 7)(2x - 12) = 0
x + 7 = 0 and 2x - 12 = 0
x = -7 and x = 6.
Since, x is given to be positive, it is 6.
So, the required consecutive numbers are 6 and 7.
Check: 6^2 + 7^2 = 36 + 49 = 85.
Good luck!
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