27x^2-512x-3=0

Solution for 27x^2-512x-3=0 equation:

27x^2-512x-3=0

a = 27; b = -512; c = -3;
Δ = b2-4ac
Δ = -5122-4·27·(-3)
Δ = 262468
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:

$\sqrt{\Delta}=\sqrt{262468}=\sqrt{4*65617}=\sqrt{4}*\sqrt{65617}=2\sqrt{65617}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-512)-2\sqrt{65617}}{2*27}=\frac{512-2\sqrt{65617}}{54}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-512)+2\sqrt{65617}}{2*27}=\frac{512+2\sqrt{65617}}{54}
$