Y=-x^2+24x-54

Solution for Y=-x^2+24x-54 equation:

=-Y^2+24Y-54

We move all terms to the left:

-(-Y^2+24Y-54)=0

We get rid of parentheses

Y^2-24Y+54=0

a = 1; b = -24; c = +54;
Δ = b2-4ac
Δ = -242-4·1·54
Δ = 360
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:

$\sqrt{\Delta}=\sqrt{360}=\sqrt{36*10}=\sqrt{36}*\sqrt{10}=6\sqrt{10}$
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-24)-6\sqrt{10}}{2*1}=\frac{24-6\sqrt{10}}{2}
$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-24)+6\sqrt{10}}{2*1}=\frac{24+6\sqrt{10}}{2}
$