(Y-7)^2=2y^2-20y+54

Solution for (Y-7)^2=2y^2-20y+54 equation:

(-7)^2=2Y^2-20Y+54

We move all terms to the left:

(-7)^2-(2Y^2-20Y+54)=0

We add all the numbers together, and all the variables

-(2Y^2-20Y+54)+49=0

We get rid of parentheses

-2Y^2+20Y-54+49=0

We add all the numbers together, and all the variables

-2Y^2+20Y-5=0

a = -2; b = 20; c = -5;
Δ = b2-4ac
Δ = 202-4·(-2)·(-5)
Δ = 360
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:

$\sqrt{\Delta}=\sqrt{360}=\sqrt{36*10}=\sqrt{36}*\sqrt{10}=6\sqrt{10}$
$Y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-6\sqrt{10}}{2*-2}=\frac{-20-6\sqrt{10}}{-4}
$
$Y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+6\sqrt{10}}{2*-2}=\frac{-20+6\sqrt{10}}{-4}
$