X^2+y^2+14y-1=0

Solution for X^2+y^2+14y-1=0 equation:

X^2+X^2+14X-1=0

We add all the numbers together, and all the variables

2X^2+14X-1=0

a = 2; b = 14; c = -1;
Δ = b2-4ac
Δ = 142-4·2·(-1)
Δ = 204
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:

$\sqrt{\Delta}=\sqrt{204}=\sqrt{4*51}=\sqrt{4}*\sqrt{51}=2\sqrt{51}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(14)-2\sqrt{51}}{2*2}=\frac{-14-2\sqrt{51}}{4}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(14)+2\sqrt{51}}{2*2}=\frac{-14+2\sqrt{51}}{4}
$