SOLUTION: Find two consecutive positive integers such that the sum of their squares is 85.

Question 102548: Find two consecutive positive integers such that the sum of
their squares is 85.

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Consecutive integers follow the form: , , , ...., etc.

Which means their squares are , , etc

So the sum of their squares is:

Foil

Subtract 85 from both sides

Let's use the quadratic formula to solve for x:

Starting with the general quadratic

the general solution using the quadratic equation is:

So lets solve ( notice , , and )

Plug in a=2, b=2, and c=-84

Square 2 to get 4

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root (note: If you need help with simplifying the square root, check out this solver)

Multiply 2 and 2 to get 4

So now the expression breaks down into two parts

or

Lets look at the first part:

Add the terms in the numerator
Divide

So one answer is

Now lets look at the second part:

Subtract the terms in the numerator
Divide

So another answer is

So our solutions are:
or

So that means our first number is either 6 or -7. If the first number is 6, then the second number is 7

Check:

Since the two squares add to 85, our answer is verified.

If the first number is -7, then the second number is -6

Check:

Since the two squares add to 85, our answer is verified.

So our two numbers could be
6 and 7

or

-7 and -6

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