If and , then what is ?

Using x + y = 2   →  y = 2 - x      and xy = 23  ...so....

x(2- x)= 23      expand

2x - x^2 = 23    rearrange

x^2 - 2x + 23 = 0    .....which implies that...

x^2 = 2x -23

And y^2 = (2 - x)^2 = (x -2)^2 = x^2 - 4x + 4

Therefore

x^2 + y^2 =

2x - 23 + x^2 - 4x + 4 =

x^2 - 2x - 19

Square both sides of the equation  x + y  = 2:

--->   (x + y)²  =  2²

Multiplying out:

--->  x² + 2xy + y²  =  4

Since xy = 23:

--->   x² + 23 + y²  =  4

--->   x² + y²  =  -19            (This does not have real answers!)

If    and   ,  then what is    ?

$$(x+y)^2=x^2+2xy+y^2 \quad | \quad -2xy\\\\ x^2+y^2 =(\underbrace{x+y}_{=2})^2-2\underbrace{xy}_{=23}\\\\ x^2+y^2 = 2^2-2*23\\\\ x^2+y^2 = 4-46\\\\ x^2+y^2 = -42$$