x*(x-1)-1/4=0

Solution for x*(x-1)-1/4=0 equation:

x(x-1)-1/4=0

We multiply parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

4x^2-4x-1=0

a = 4; b = -4; c = -1;
Δ = b2-4ac
Δ = -42-4·4·(-1)
Δ = 32
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:

$\sqrt{\Delta}=\sqrt{32}=\sqrt{16*2}=\sqrt{16}*\sqrt{2}=4\sqrt{2}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-4\sqrt{2}}{2*4}=\frac{4-4\sqrt{2}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+4\sqrt{2}}{2*4}=\frac{4+4\sqrt{2}}{8}
$