Y=12x^2-120x+209

Solution for Y=12x^2-120x+209 equation:

=12Y^2-120Y+209

We move all terms to the left:

-(12Y^2-120Y+209)=0

We get rid of parentheses

-12Y^2+120Y-209=0

a = -12; b = 120; c = -209;
Δ = b2-4ac
Δ = 1202-4·(-12)·(-209)
Δ = 4368
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:

$\sqrt{\Delta}=\sqrt{4368}=\sqrt{16*273}=\sqrt{16}*\sqrt{273}=4\sqrt{273}$
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(120)-4\sqrt{273}}{2*-12}=\frac{-120-4\sqrt{273}}{-24}
$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(120)+4\sqrt{273}}{2*-12}=\frac{-120+4\sqrt{273}}{-24}
$