9x^2=3x+1

Solution for 9x^2=3x+1 equation:

9x^2=3x+1

We move all terms to the left:

9x^2-(3x+1)=0

We get rid of parentheses

9x^2-3x-1=0

a = 9; b = -3; c = -1;
Δ = b2-4ac
Δ = -32-4·9·(-1)
Δ = 45
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:

$\sqrt{\Delta}=\sqrt{45}=\sqrt{9*5}=\sqrt{9}*\sqrt{5}=3\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-3\sqrt{5}}{2*9}=\frac{3-3\sqrt{5}}{18}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+3\sqrt{5}}{2*9}=\frac{3+3\sqrt{5}}{18}
$