x^2-x=6.5

Solution for x^2-x=6.5 equation:

x^2-x=6.5

We move all terms to the left:

x^2-x-(6.5)=0

We add all the numbers together, and all the variables

x^2-1x-6.5=0

a = 1; b = -1; c = -6.5;
Δ = b2-4ac
Δ = -12-4·1·(-6.5)
Δ = 27
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:

$\sqrt{\Delta}=\sqrt{27}=\sqrt{9*3}=\sqrt{9}*\sqrt{3}=3\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-3\sqrt{3}}{2*1}=\frac{1-3\sqrt{3}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+3\sqrt{3}}{2*1}=\frac{1+3\sqrt{3}}{2}
$