1/2x+3x=2x+42

Solution for 1/2x+3x=2x+42 equation:

1/2x+3x=2x+42

We move all terms to the left:

1/2x+3x-(2x+42)=0


Domain of the equation: 2x!=0

x!=0/2

x!=0

x∈R


We add all the numbers together, and all the variables

3x+1/2x-(2x+42)=0

We get rid of parentheses

3x+1/2x-2x-42=0

We multiply all the terms by the denominator


3x*2x-2x*2x-42*2x+1=0

Wy multiply elements

6x^2-4x^2-84x+1=0

We add all the numbers together, and all the variables

2x^2-84x+1=0

a = 2; b = -84; c = +1;
Δ = b2-4ac
Δ = -842-4·2·1
Δ = 7048
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:

$\sqrt{\Delta}=\sqrt{7048}=\sqrt{4*1762}=\sqrt{4}*\sqrt{1762}=2\sqrt{1762}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-84)-2\sqrt{1762}}{2*2}=\frac{84-2\sqrt{1762}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-84)+2\sqrt{1762}}{2*2}=\frac{84+2\sqrt{1762}}{4}
$