X^2-6x^2-9x+54=0

Solution for X^2-6x^2-9x+54=0 equation:

X^2-6X^2-9X+54=0

We add all the numbers together, and all the variables

-5X^2-9X+54=0

a = -5; b = -9; c = +54;
Δ = b2-4ac
Δ = -92-4·(-5)·54
Δ = 1161
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:

$\sqrt{\Delta}=\sqrt{1161}=\sqrt{9*129}=\sqrt{9}*\sqrt{129}=3\sqrt{129}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-3\sqrt{129}}{2*-5}=\frac{9-3\sqrt{129}}{-10}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+3\sqrt{129}}{2*-5}=\frac{9+3\sqrt{129}}{-10}
$