3x-1/2x=9+x

Solution for 3x-1/2x=9+x equation:

3x-1/2x=9+x

We move all terms to the left:

3x-1/2x-(9+x)=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-(x+9)=0

We get rid of parentheses

3x-1/2x-x-9=0

We multiply all the terms by the denominator


3x*2x-x*2x-9*2x-1=0

Wy multiply elements

6x^2-2x^2-18x-1=0

We add all the numbers together, and all the variables

4x^2-18x-1=0

a = 4; b = -18; c = -1;
Δ = b2-4ac
Δ = -182-4·4·(-1)
Δ = 340
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:

$\sqrt{\Delta}=\sqrt{340}=\sqrt{4*85}=\sqrt{4}*\sqrt{85}=2\sqrt{85}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-18)-2\sqrt{85}}{2*4}=\frac{18-2\sqrt{85}}{8}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-18)+2\sqrt{85}}{2*4}=\frac{18+2\sqrt{85}}{8}
$