9y(2y-3)=5(y-2)+2y

Solution for 9y(2y-3)=5(y-2)+2y equation:

9y(2y-3)=5(y-2)+2y

We move all terms to the left:

9y(2y-3)-(5(y-2)+2y)=0

We multiply parentheses

18y^2-27y-(5(y-2)+2y)=0


We calculate terms in parentheses: -(5(y-2)+2y), so:

5(y-2)+2y

We add all the numbers together, and all the variables

2y+5(y-2)

We multiply parentheses

2y+5y-10

We add all the numbers together, and all the variables

7y-10

Back to the equation:

-(7y-10)


We get rid of parentheses

18y^2-27y-7y+10=0

We add all the numbers together, and all the variables

18y^2-34y+10=0

a = 18; b = -34; c = +10;
Δ = b2-4ac
Δ = -342-4·18·10
Δ = 436
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:

$\sqrt{\Delta}=\sqrt{436}=\sqrt{4*109}=\sqrt{4}*\sqrt{109}=2\sqrt{109}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-34)-2\sqrt{109}}{2*18}=\frac{34-2\sqrt{109}}{36}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-34)+2\sqrt{109}}{2*18}=\frac{34+2\sqrt{109}}{36}
$