X(x-1)+(x-2)(x-3)=42

Solution for X(x-1)+(x-2)(x-3)=42 equation:

X(X-1)+(X-2)(X-3)=42

We move all terms to the left:

X(X-1)+(X-2)(X-3)-(42)=0

We multiply parentheses

X^2-1X+(X-2)(X-3)-42=0

We multiply parentheses ..

X^2+(+X^2-3X-2X+6)-1X-42=0

We get rid of parentheses

X^2+X^2-3X-2X-1X+6-42=0

We add all the numbers together, and all the variables

2X^2-6X-36=0

a = 2; b = -6; c = -36;
Δ = b2-4ac
Δ = -62-4·2·(-36)
Δ = 324
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{324}=18$
$X_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-6)-18}{2*2}=\frac{-12}{4}
=-3
$
$X_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-6)+18}{2*2}=\frac{24}{4}
=6
$