(Y-3)^2=2y^2-3y-19

Solution for (Y-3)^2=2y^2-3y-19 equation:

(-3)^2=2Y^2-3Y-19

We move all terms to the left:

(-3)^2-(2Y^2-3Y-19)=0

We add all the numbers together, and all the variables

-(2Y^2-3Y-19)+9=0

We get rid of parentheses

-2Y^2+3Y+19+9=0

We add all the numbers together, and all the variables

-2Y^2+3Y+28=0

a = -2; b = 3; c = +28;
Δ = b2-4ac
Δ = 32-4·(-2)·28
Δ = 233
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(3)-\sqrt{233}}{2*-2}=\frac{-3-\sqrt{233}}{-4}
$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(3)+\sqrt{233}}{2*-2}=\frac{-3+\sqrt{233}}{-4}
$