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

Question 93439: Find to consecutive integers such that the sum of their squares is 85.
Found 2 solutions by checkley75, jim_thompson5910:
Answer by checkley75(3666) (Show Source): You can put this solution on YOUR website!
x^2+(x+1)^2=85
x^2+x^2+2x+1=85
2x^2+2x+1-85=0
2x^2+2x-84=0
2(x^2+x-42)=0
2(x+7)(x-6)=0
x-6=0
x=6 answer.
x+1=7 answer.
proof
6^2+7^2=85
36+49=85
85=85

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Start with the given equation

Foil

Combine like terms

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

Since we only care about the positive numbers, our solution is:

So our second number is

which means our numbers are

6 and 7
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