Question 144981
"The sum of two numbers is 8" translates to {{{x+y=8}}}



"The sum of the squares of the two numbers is 34" translates to {{{x^2+y^2=34}}}



{{{x+y=8}}} Start with the first equation



{{{y=8-x}}} Subtract x from both sides to isolate y



{{{x^2+y^2=34}}} Move onto the second equation



{{{x^2+(8-x)^2=34}}} Plug in {{{y=8-x}}}



{{{x^2+64-16x+x^2=34}}} Foil



{{{x^2+64-16x+x^2-34=0}}} Get all terms to the left side.



{{{2x^2-16x+30=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=2}}}, {{{b=-16}}}, and {{{c=30}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(-16) +- sqrt( (-16)^2-4(2)(30) ))/(2(2))}}} Plug in  {{{a=2}}}, {{{b=-16}}}, and {{{c=30}}}



{{{x = (16 +- sqrt( (-16)^2-4(2)(30) ))/(2(2))}}} Negate {{{-16}}} to get {{{16}}}. 



{{{x = (16 +- sqrt( 256-4(2)(30) ))/(2(2))}}} Square {{{-16}}} to get {{{256}}}. 



{{{x = (16 +- sqrt( 256-240 ))/(2(2))}}} Multiply {{{4(2)(30)}}} to get {{{240}}}



{{{x = (16 +- sqrt( 16 ))/(2(2))}}} Subtract {{{240}}} from {{{256}}} to get {{{16}}}



{{{x = (16 +- sqrt( 16 ))/(4)}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}. 



{{{x = (16 +- 4)/(4)}}} Take the square root of {{{16}}} to get {{{4}}}. 



{{{x = (16 + 4)/(4)}}} or {{{x = (16 - 4)/(4)}}} Break up the expression. 



{{{x = (20)/(4)}}} or {{{x =  (12)/(4)}}} Combine like terms. 



{{{x = 5}}} or {{{x = 3}}} Simplify. 



So our answers are {{{x = 5}}} or {{{x = 3}}}