Question 95825
X+Y=12 OR X=12-Y
X^2+Y^2=78 
(12-Y)^2+Y^2=78
144-24Y+Y^2+Y^2=78
2Y^2-24Y+144-78=0
2Y^2-24Y+66=0
2(Y^2-12Y+33)=0
USING THE QUADRATIC EQUATION YOU ARE CORRECT:
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 
X=(12+-SQRT[-12^2-4*1*33])/2*1
X=(12+-SQRT[144-132])/2
X=(12+-SQRT[12])/2
X=(12+-3.464)/2
X=(12+3.464)/2
X=15.464/2
X=7.73 ANSWER.
X=(12-3.464)/2
X=8.536/2
X=4.27 ANSWER.
PROOF
7.73+4.27=12
12=12
7.73^2+4.27^2=78
59.7529+18.2329=78
78=78