x^2-4x-1/3=0

Solution for x^2-4x-1/3=0 equation:

x^2-4x-1/3=0

We multiply all the terms by the denominator


x^2*3-4x*3-1=0

Wy multiply elements

3x^2-12x-1=0

a = 3; b = -12; c = -1;
Δ = b2-4ac
Δ = -122-4·3·(-1)
Δ = 156
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:

$\sqrt{\Delta}=\sqrt{156}=\sqrt{4*39}=\sqrt{4}*\sqrt{39}=2\sqrt{39}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{39}}{2*3}=\frac{12-2\sqrt{39}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{39}}{2*3}=\frac{12+2\sqrt{39}}{6}
$