Y=12x-x^2+3

Solution for Y=12x-x^2+3 equation:

=12Y-Y^2+3

We move all terms to the left:

-(12Y-Y^2+3)=0

We get rid of parentheses

Y^2-12Y-3=0

a = 1; b = -12; c = -3;
Δ = b2-4ac
Δ = -122-4·1·(-3)
Δ = 156
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:

$\sqrt{\Delta}=\sqrt{156}=\sqrt{4*39}=\sqrt{4}*\sqrt{39}=2\sqrt{39}$
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{39}}{2*1}=\frac{12-2\sqrt{39}}{2}
$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{39}}{2*1}=\frac{12+2\sqrt{39}}{2}
$