2x(x-9)=3(x-6)

Solution for 2x(x-9)=3(x-6) equation:

2x(x-9)=3(x-6)

We move all terms to the left:

2x(x-9)-(3(x-6))=0

We multiply parentheses

2x^2-18x-(3(x-6))=0


We calculate terms in parentheses: -(3(x-6)), so:

3(x-6)

We multiply parentheses

3x-18

Back to the equation:

-(3x-18)


We get rid of parentheses

2x^2-18x-3x+18=0

We add all the numbers together, and all the variables

2x^2-21x+18=0

a = 2; b = -21; c = +18;
Δ = b2-4ac
Δ = -212-4·2·18
Δ = 297
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{297}=\sqrt{9*33}=\sqrt{9}*\sqrt{33}=3\sqrt{33}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-21)-3\sqrt{33}}{2*2}=\frac{21-3\sqrt{33}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-21)+3\sqrt{33}}{2*2}=\frac{21+3\sqrt{33}}{4}
$