1/2x+100=3x-400

Solution for 1/2x+100=3x-400 equation:

1/2x+100=3x-400

We move all terms to the left:

1/2x+100-(3x-400)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We get rid of parentheses

1/2x-3x+400+100=0

We multiply all the terms by the denominator


-3x*2x+400*2x+100*2x+1=0

Wy multiply elements

-6x^2+800x+200x+1=0

We add all the numbers together, and all the variables

-6x^2+1000x+1=0

a = -6; b = 1000; c = +1;
Δ = b2-4ac
Δ = 10002-4·(-6)·1
Δ = 1000024
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:

$\sqrt{\Delta}=\sqrt{1000024}=\sqrt{4*250006}=\sqrt{4}*\sqrt{250006}=2\sqrt{250006}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(1000)-2\sqrt{250006}}{2*-6}=\frac{-1000-2\sqrt{250006}}{-12}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(1000)+2\sqrt{250006}}{2*-6}=\frac{-1000+2\sqrt{250006}}{-12}
$