x^2-x-1=289

Solution for x^2-x-1=289 equation:

x^2-x-1=289

We move all terms to the left:

x^2-x-1-(289)=0

We add all the numbers together, and all the variables

x^2-1x-290=0

a = 1; b = -1; c = -290;
Δ = b2-4ac
Δ = -12-4·1·(-290)
Δ = 1161
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:

$\sqrt{\Delta}=\sqrt{1161}=\sqrt{9*129}=\sqrt{9}*\sqrt{129}=3\sqrt{129}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-1)-3\sqrt{129}}{2*1}=\frac{1-3\sqrt{129}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-1)+3\sqrt{129}}{2*1}=\frac{1+3\sqrt{129}}{2}
$