Y=0.5x^2+2x-6

Solution for Y=0.5x^2+2x-6 equation:

=0.5Y^2+2Y-6

We move all terms to the left:

-(0.5Y^2+2Y-6)=0

We get rid of parentheses

-0.5Y^2-2Y+6=0

a = -0.5; b = -2; c = +6;
Δ = b2-4ac
Δ = -22-4·(-0.5)·6
Δ = 16
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{16}=4$
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-4}{2*-0.5}=\frac{-2}{-1}
=+2
$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+4}{2*-0.5}=\frac{6}{-1}
=-6
$