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

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Consecutive integers follow the form: , , , ...., etc.


Which means their squares are , , etc

So the sum of their squares is:




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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