(6+x)/x=x/2+3

Solution for (6+x)/x=x/2+3 equation:

(6+x)/x=x/2+3

We move all terms to the left:

(6+x)/x-(x/2+3)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

(x+6)/x-(x/2+3)=0

We get rid of parentheses

(x+6)/x-x/2-3=0

We calculate fractions


(-1x^2)/2x+(2x+12)/2x-3=0

We multiply all the terms by the denominator


(-1x^2)+(2x+12)-3*2x=0

Wy multiply elements

(-1x^2)+(2x+12)-6x=0

We get rid of parentheses

-1x^2+2x-6x+12=0

We add all the numbers together, and all the variables

-1x^2-4x+12=0

a = -1; b = -4; c = +12;
Δ = b2-4ac
Δ = -42-4·(-1)·12
Δ = 64
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{64}=8$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-8}{2*-1}=\frac{-4}{-2}
=+2
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+8}{2*-1}=\frac{12}{-2}
=-6
$