1/2*x+20=3x-85

Solution for 1/2*x+20=3x-85 equation:

1/2x+20=3x-85

We move all terms to the left:

1/2x+20-(3x-85)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-3x+85+20=0

We multiply all the terms by the denominator


-3x*2x+85*2x+20*2x+1=0

Wy multiply elements

-6x^2+170x+40x+1=0

We add all the numbers together, and all the variables

-6x^2+210x+1=0

a = -6; b = 210; c = +1;
Δ = b2-4ac
Δ = 2102-4·(-6)·1
Δ = 44124
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:

$\sqrt{\Delta}=\sqrt{44124}=\sqrt{4*11031}=\sqrt{4}*\sqrt{11031}=2\sqrt{11031}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(210)-2\sqrt{11031}}{2*-6}=\frac{-210-2\sqrt{11031}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(210)+2\sqrt{11031}}{2*-6}=\frac{-210+2\sqrt{11031}}{-12}
$