Y=-3x^2+8x+9

Solution for Y=-3x^2+8x+9 equation:

=-3Y^2+8Y+9

We move all terms to the left:

-(-3Y^2+8Y+9)=0

We get rid of parentheses

3Y^2-8Y-9=0

a = 3; b = -8; c = -9;
Δ = b2-4ac
Δ = -82-4·3·(-9)
Δ = 172
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:

$\sqrt{\Delta}=\sqrt{172}=\sqrt{4*43}=\sqrt{4}*\sqrt{43}=2\sqrt{43}$
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-8)-2\sqrt{43}}{2*3}=\frac{8-2\sqrt{43}}{6}
$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-8)+2\sqrt{43}}{2*3}=\frac{8+2\sqrt{43}}{6}
$