(x-1)(x-2)=1

Solution for (x-1)(x-2)=1 equation:

(x-1)(x-2)=1

We move all terms to the left:

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

We multiply parentheses ..

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

We get rid of parentheses

x^2-2x-1x+2-1=0

We add all the numbers together, and all the variables

x^2-3x+1=0

a = 1; b = -3; c = +1;
Δ = b2-4ac
Δ = -32-4·1·1
Δ = 5
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-3)-\sqrt{5}}{2*1}=\frac{3-\sqrt{5}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-3)+\sqrt{5}}{2*1}=\frac{3+\sqrt{5}}{2}
$