Question 102548
Consecutive integers follow the form: {{{x}}}, {{{x+1}}}, {{{x+2}}}, ...., etc.



Which means their squares are {{{x^2}}}, {{{(x+1)^2}}}, etc


So the sum of their squares is:


{{{x^2+(x+1)^2=85}}}



{{{x^2+x^2+2x+1=85}}} Foil



{{{2x^2+2x-84=0}}} Subtract 85 from both sides


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



Starting with the general quadratic


{{{ax^2+bx+c=0}}}


the general solution using the quadratic equation is:


{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}




So lets solve {{{2*x^2+2*x-84=0}}} ( notice {{{a=2}}}, {{{b=2}}}, and {{{c=-84}}})





{{{x = (-2 +- sqrt( (2)^2-4*2*-84 ))/(2*2)}}} Plug in a=2, b=2, and c=-84




{{{x = (-2 +- sqrt( 4-4*2*-84 ))/(2*2)}}} Square 2 to get 4  




{{{x = (-2 +- sqrt( 4+672 ))/(2*2)}}} Multiply {{{-4*-84*2}}} to get {{{672}}}




{{{x = (-2 +- sqrt( 676 ))/(2*2)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (-2 +- 26)/(2*2)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{x = (-2 +- 26)/4}}} Multiply 2 and 2 to get 4


So now the expression breaks down into two parts


{{{x = (-2 + 26)/4}}} or {{{x = (-2 - 26)/4}}}


Lets look at the first part:


{{{x=(-2 + 26)/4}}}


{{{x=24/4}}} Add the terms in the numerator

{{{x=6}}} Divide


So one answer is

{{{x=6}}}




Now lets look at the second part:


{{{x=(-2 - 26)/4}}}


{{{x=-28/4}}} Subtract the terms in the numerator

{{{x=-7}}} Divide


So another answer is

{{{x=-7}}}


So our solutions are:

{{{x=6}}} or {{{x=-7}}}



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



Check:


{{{6^2+7^2=36+49=85}}} Since the two squares add to 85, our answer is verified.


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



Check:


{{{(-7)^2+(-6)^2=49+36=85}}} Since the two squares add to 85, our answer is verified.




So our two numbers could be

6 and 7


or 


-7 and -6