9(5q+6)q=1

Solution for 9(5q+6)q=1 equation:

9(5q+6)q=1

We move all terms to the left:

9(5q+6)q-(1)=0

We multiply parentheses

45q^2+54q-1=0

a = 45; b = 54; c = -1;
Δ = b2-4ac
Δ = 542-4·45·(-1)
Δ = 3096
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{3096}=\sqrt{36*86}=\sqrt{36}*\sqrt{86}=6\sqrt{86}$
$q_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(54)-6\sqrt{86}}{2*45}=\frac{-54-6\sqrt{86}}{90}
$
$q_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(54)+6\sqrt{86}}{2*45}=\frac{-54+6\sqrt{86}}{90}
$