(x-1)(x-4)=1

Solution for (x-1)(x-4)=1 equation:

(x-1)(x-4)=1

We move all terms to the left:

(x-1)(x-4)-(1)=0

We multiply parentheses ..

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

We get rid of parentheses

x^2-4x-1x+4-1=0

We add all the numbers together, and all the variables

x^2-5x+3=0

a = 1; b = -5; c = +3;
Δ = b2-4ac
Δ = -52-4·1·3
Δ = 13
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{-(-5)-\sqrt{13}}{2*1}=\frac{5-\sqrt{13}}{2}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-5)+\sqrt{13}}{2*1}=\frac{5+\sqrt{13}}{2}
$