Y2+x2+8x=84

Solution for Y2+x2+8x=84 equation:

2+Y2+8Y=84

We move all terms to the left:

2+Y2+8Y-(84)=0

We add all the numbers together, and all the variables

Y^2+8Y-82=0

a = 1; b = 8; c = -82;
Δ = b2-4ac
Δ = 82-4·1·(-82)
Δ = 392
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:

$\sqrt{\Delta}=\sqrt{392}=\sqrt{196*2}=\sqrt{196}*\sqrt{2}=14\sqrt{2}$
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-14\sqrt{2}}{2*1}=\frac{-8-14\sqrt{2}}{2}
$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+14\sqrt{2}}{2*1}=\frac{-8+14\sqrt{2}}{2}
$