r^2=4r+1=0

Solution for r^2=4r+1=0 equation:

r^2=4r+1=0

We move all terms to the left:

r^2-(4r+1)=0

We get rid of parentheses

r^2-4r-1=0

a = 1; b = -4; c = -1;
Δ = b2-4ac
Δ = -42-4·1·(-1)
Δ = 20
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{20}=\sqrt{4*5}=\sqrt{4}*\sqrt{5}=2\sqrt{5}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{5}}{2*1}=\frac{4-2\sqrt{5}}{2}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{5}}{2*1}=\frac{4+2\sqrt{5}}{2}
$