X^2+y^2=-10y-9

Solution for X^2+y^2=-10y-9 equation:

X^2+X^2=-10X-9

We move all terms to the left:

X^2+X^2-(-10X-9)=0

We add all the numbers together, and all the variables

2X^2-(-10X-9)=0

We get rid of parentheses

2X^2+10X+9=0

a = 2; b = 10; c = +9;
Δ = b2-4ac
Δ = 102-4·2·9
Δ = 28
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:

$\sqrt{\Delta}=\sqrt{28}=\sqrt{4*7}=\sqrt{4}*\sqrt{7}=2\sqrt{7}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{7}}{2*2}=\frac{-10-2\sqrt{7}}{4}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{7}}{2*2}=\frac{-10+2\sqrt{7}}{4}
$