Y=-x^2+6x+6

Solution for Y=-x^2+6x+6 equation:

=-Y^2+6Y+6

We move all terms to the left:

-(-Y^2+6Y+6)=0

We get rid of parentheses

Y^2-6Y-6=0

a = 1; b = -6; c = -6;
Δ = b2-4ac
Δ = -62-4·1·(-6)
Δ = 60
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:

$\sqrt{\Delta}=\sqrt{60}=\sqrt{4*15}=\sqrt{4}*\sqrt{15}=2\sqrt{15}$
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-2\sqrt{15}}{2*1}=\frac{6-2\sqrt{15}}{2}
$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+2\sqrt{15}}{2*1}=\frac{6+2\sqrt{15}}{2}
$