9x^2=12x-1

Solution for 9x^2=12x-1 equation:

9x^2=12x-1

We move all terms to the left:

9x^2-(12x-1)=0

We get rid of parentheses

9x^2-12x+1=0

a = 9; b = -12; c = +1;
Δ = b2-4ac
Δ = -122-4·9·1
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-6\sqrt{3}}{2*9}=\frac{12-6\sqrt{3}}{18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+6\sqrt{3}}{2*9}=\frac{12+6\sqrt{3}}{18}
$