X^2+y^2+10y-23=0

Solution for X^2+y^2+10y-23=0 equation:

X^2+X^2+10X-23=0

We add all the numbers together, and all the variables

2X^2+10X-23=0

a = 2; b = 10; c = -23;
Δ = b2-4ac
Δ = 102-4·2·(-23)
Δ = 284
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{284}=\sqrt{4*71}=\sqrt{4}*\sqrt{71}=2\sqrt{71}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{71}}{2*2}=\frac{-10-2\sqrt{71}}{4}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{71}}{2*2}=\frac{-10+2\sqrt{71}}{4}
$