Question 376333
"The sum of the squares of two consecutive integers is 13" means that {{{x^2+(x+1)^2=13}}}



{{{x^2+(x+1)^2=13}}} Start with the given equation.



{{{x^2+x^2+2x+1=13}}} FOIL



{{{x^2+x^2+2x+1-13=0}}} Subtract 13 from both sides.



{{{2x^2+2x-12=0}}} Combine like terms.



{{{2(x^2+x-6)=0}}} Factor out the GCF 2



{{{2(x+3)(x-2)=0}}} Factor {{{x^2+x-6}}} to get {{{(x+3)(x-2)}}}



{{{x+3=0}}} or {{{x-2=0}}} Use the zero product property



{{{x=-3}}} or {{{x=2}}} Solve for x.



So if {{{x=2}}}, then {{{x+1=2+1=3}}} making the two numbers to be 2 and 3. Notice how {{{2^2+3^2=4+9=13}}}



Or, if {{{x=-3}}}, then {{{x+1=-3+1=-2}}} making the two numbers to be -3 and 2. Notice how {{{(-3)^2+(-2)^2=9+4=13}}}



So the two numbers are


2 and 3


OR


-3 and -2



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Jim