Question 177254
"Two numbers differ by 6" translates to {{{x-y=6}}}. If we solve for "y", we get {{{y=x-6}}}


"if the numbers are squared and then added, the result is 146" means that {{{x^2+y^2=146}}}



{{{x^2+y^2=146}}} Start with the second equation.



{{{x^2+(x-6)^2=146}}} Plug in {{{y=x-6}}}



{{{x^2+x^2-12x+36=146}}} FOIL



{{{x^2+x^2-12x+36-146=0}}} Subtract 146 from both sides.



{{{2x^2-12x-110=0}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=2}}}, {{{b=-12}}}, and {{{c=-110}}}



Let's use the quadratic formula to solve for x



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



{{{x = (-(-12) +- sqrt( (-12)^2-4(2)(-110) ))/(2(2))}}} Plug in  {{{a=2}}}, {{{b=-12}}}, and {{{c=-110}}}



{{{x = (12 +- sqrt( (-12)^2-4(2)(-110) ))/(2(2))}}} Negate {{{-12}}} to get {{{12}}}. 



{{{x = (12 +- sqrt( 144-4(2)(-110) ))/(2(2))}}} Square {{{-12}}} to get {{{144}}}. 



{{{x = (12 +- sqrt( 144--880 ))/(2(2))}}} Multiply {{{4(2)(-110)}}} to get {{{-880}}}



{{{x = (12 +- sqrt( 144+880 ))/(2(2))}}} Rewrite {{{sqrt(144--880)}}} as {{{sqrt(144+880)}}}



{{{x = (12 +- sqrt( 1024 ))/(2(2))}}} Add {{{144}}} to {{{880}}} to get {{{1024}}}



{{{x = (12 +- sqrt( 1024 ))/(4)}}} Multiply {{{2}}} and {{{2}}} to get {{{4}}}. 



{{{x = (12 +- 32)/(4)}}} Take the square root of {{{1024}}} to get {{{32}}}. 



{{{x = (12 + 32)/(4)}}} or {{{x = (12 - 32)/(4)}}} Break up the expression. 



{{{x = (44)/(4)}}} or {{{x =  (-20)/(4)}}} Combine like terms. 



{{{x = 11}}} or {{{x = -5}}} Simplify. 



So the first numbers are either {{{x = 11}}} or {{{x = -5}}} 



{{{y=x-6}}} Go back to the isolated equation



{{{y=11-6}}} Plug in the first solution {{{x=11}}}



{{{y=5}}} Subtract



So when {{{x = 11}}}, then {{{y=5}}}


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{{{y=x-6}}} Go back to the isolated equation



{{{y=-5-6}}} Plug in the first solution {{{x=11}}}



{{{y=-11}}} Subtract



So when {{{x = -5}}}, then {{{y=-11}}}



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Answer:



So this means that the two numbers are


11 and 5


OR


-5 and -11