Question 10603
let 2 numbers be x and y.


{{{(x+y)^2 = 20 + (x-y)^2}}} --eqn1
also, {{{x^2 - y^2 = 24}}} --eqn2


eqn1 becomes {{{x^2 + 2xy + y^2 = 20 + x^2 - y^2}}}
{{{2xy + 2y^2 = 20}}}
{{{xy + y^2 = 10}}}
{{{xy = 10 - y^2}}}... now square both sides
so, {{{(xy)^2 = (10 - y^2)^2}}}
which is {{{x^2y^2 = 100 - 20y^2 + y^4}}}


eqn2 becomes {{{x^2 - y^2 = 24}}}
{{{x^2 = y^2 + 24}}}


Sub this into eqn1, to give
{{{(y^2 + 24)y^2 = 100 - 20y^2 + y^4}}}
{{{y^4 + 24y^2 = 100 - 20y^2 + y^4}}}
{{{44y^2 = 100}}}
{{{y^2 = 100/44}}}
{{{y^2 = 25/11}}}
{{{y = -sqrt(25/11)}}} ..just take the negative version, since the question asked for that.
{{{y = -5/sqrt(11)}}}


now, {{{x^2 = y^2 + 24}}}, so {{{x^2 = (25/11) + 24}}}
--> x^2 = 289/11
--> {{{x = -sqrt(289/11))}}}
--> {{{x = -17/sqrt(11))}}}


check:
{{{(x+y)^2}}} is 44
{{{(x-y)^2}}} is 13.09
so this is wrong!, since it should differ by 20.


and {{{x^2 - y^2}}} is 289/11 - 25/11 which is 24... correct.


so there might be something i am glaringly missing here...check my working: the method is correct!


jon.