y^2=40-X^2

Solution for y^2=40-X^2 equation:

y^2=40-y^2

We move all terms to the left:

y^2-(40-y^2)=0

We get rid of parentheses

y^2+y^2-40=0

We add all the numbers together, and all the variables

2y^2-40=0

a = 2; b = 0; c = -40;
Δ = b2-4ac
Δ = 02-4·2·(-40)
Δ = 320
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{320}=\sqrt{64*5}=\sqrt{64}*\sqrt{5}=8\sqrt{5}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-8\sqrt{5}}{2*2}=\frac{0-8\sqrt{5}}{4}
=-\frac{8\sqrt{5}}{4}
=-2\sqrt{5}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+8\sqrt{5}}{2*2}=\frac{0+8\sqrt{5}}{4}
=\frac{8\sqrt{5}}{4}
=2\sqrt{5}
$