5n^2-5n=1

Solution for 5n^2-5n=1 equation:

5n^2-5n=1

We move all terms to the left:

5n^2-5n-(1)=0

a = 5; b = -5; c = -1;
Δ = b2-4ac
Δ = -52-4·5·(-1)
Δ = 45
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{45}=\sqrt{9*5}=\sqrt{9}*\sqrt{5}=3\sqrt{5}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-5)-3\sqrt{5}}{2*5}=\frac{5-3\sqrt{5}}{10}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-5)+3\sqrt{5}}{2*5}=\frac{5+3\sqrt{5}}{10}
$