(2x+3)(x-2)=(4-x)(5x-2x)+1

Solution for (2x+3)(x-2)=(4-x)(5x-2x)+1 equation:

(2x+3)(x-2)=(4-x)(5x-2x)+1

We move all terms to the left:

(2x+3)(x-2)-((4-x)(5x-2x)+1)=0

We add all the numbers together, and all the variables

(2x+3)(x-2)-((-1x+4)(+3x)+1)=0

We multiply parentheses ..

(+2x^2-4x+3x-6)-((-1x+4)(+3x)+1)=0


We calculate terms in parentheses: -((-1x+4)(+3x)+1), so:

(-1x+4)(+3x)+1

We multiply parentheses ..

(-3x^2+12x)+1

We get rid of parentheses

-3x^2+12x+1

Back to the equation:

-(-3x^2+12x+1)


We get rid of parentheses

2x^2+3x^2-4x+3x-12x-6-1=0

We add all the numbers together, and all the variables

5x^2-13x-7=0

a = 5; b = -13; c = -7;
Δ = b2-4ac
Δ = -132-4·5·(-7)
Δ = 309
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{-(-13)-\sqrt{309}}{2*5}=\frac{13-\sqrt{309}}{10}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-13)+\sqrt{309}}{2*5}=\frac{13+\sqrt{309}}{10}
$