X*x+1=x+x+1+41

Solution for X*x+1=x+x+1+41 equation:

X*X+1=X+X+1+41

We move all terms to the left:

X*X+1-(X+X+1+41)=0

We add all the numbers together, and all the variables

X*X-(2X+42)+1=0

Wy multiply elements

X^2-(2X+42)+1=0

We get rid of parentheses

X^2-2X-42+1=0

We add all the numbers together, and all the variables

X^2-2X-41=0

a = 1; b = -2; c = -41;
Δ = b2-4ac
Δ = -22-4·1·(-41)
Δ = 168
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:

$\sqrt{\Delta}=\sqrt{168}=\sqrt{4*42}=\sqrt{4}*\sqrt{42}=2\sqrt{42}$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{42}}{2*1}=\frac{2-2\sqrt{42}}{2}
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{42}}{2*1}=\frac{2+2\sqrt{42}}{2}
$