Y=1.5x^2-6x+3

Solution for Y=1.5x^2-6x+3 equation:

=1.5Y^2-6Y+3

We move all terms to the left:

-(1.5Y^2-6Y+3)=0

We get rid of parentheses

-1.5Y^2+6Y-3=0

a = -1.5; b = 6; c = -3;
Δ = b2-4ac
Δ = 62-4·(-1.5)·(-3)
Δ = 18
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:

$\sqrt{\Delta}=\sqrt{18}=\sqrt{9*2}=\sqrt{9}*\sqrt{2}=3\sqrt{2}$
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(6)-3\sqrt{2}}{2*-1.5}=\frac{-6-3\sqrt{2}}{-3}
$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(6)+3\sqrt{2}}{2*-1.5}=\frac{-6+3\sqrt{2}}{-3}
$