x(2x-3)-(5+x)=83

Solution for x(2x-3)-(5+x)=83 equation:

x(2x-3)-(5+x)=83

We move all terms to the left:

x(2x-3)-(5+x)-(83)=0

We add all the numbers together, and all the variables

x(2x-3)-(x+5)-83=0

We multiply parentheses

2x^2-3x-(x+5)-83=0

We get rid of parentheses

2x^2-3x-x-5-83=0

We add all the numbers together, and all the variables

2x^2-4x-88=0

a = 2; b = -4; c = -88;
Δ = b2-4ac
Δ = -42-4·2·(-88)
Δ = 720
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:

$\sqrt{\Delta}=\sqrt{720}=\sqrt{144*5}=\sqrt{144}*\sqrt{5}=12\sqrt{5}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-12\sqrt{5}}{2*2}=\frac{4-12\sqrt{5}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+12\sqrt{5}}{2*2}=\frac{4+12\sqrt{5}}{4}
$