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

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

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

We move all terms to the left:

2x(x-2)+3x(x-3)-(1)=0

We multiply parentheses

2x^2+3x^2-4x-9x-1=0

We add all the numbers together, and all the variables

5x^2-13x-1=0

a = 5; b = -13; c = -1;
Δ = b2-4ac
Δ = -132-4·5·(-1)
Δ = 189
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:

$\sqrt{\Delta}=\sqrt{189}=\sqrt{9*21}=\sqrt{9}*\sqrt{21}=3\sqrt{21}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-13)-3\sqrt{21}}{2*5}=\frac{13-3\sqrt{21}}{10}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-13)+3\sqrt{21}}{2*5}=\frac{13+3\sqrt{21}}{10}
$