x/3x+4=x+3/3x+19

Solution for x/3x+4=x+3/3x+19 equation:

x/3x+4=x+3/3x+19

We move all terms to the left:

x/3x+4-(x+3/3x+19)=0


Domain of the equation: 3x!=0

x!=0/3

x!=0

x∈R



Domain of the equation: 3x+19)!=0

x∈R


We get rid of parentheses

x/3x-x-3/3x-19+4=0

We multiply all the terms by the denominator


x-x*3x-19*3x+4*3x-3=0

Wy multiply elements

-3x^2+x-57x+12x-3=0

We add all the numbers together, and all the variables

-3x^2-44x-3=0

a = -3; b = -44; c = -3;
Δ = b2-4ac
Δ = -442-4·(-3)·(-3)
Δ = 1900
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:

$\sqrt{\Delta}=\sqrt{1900}=\sqrt{100*19}=\sqrt{100}*\sqrt{19}=10\sqrt{19}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-44)-10\sqrt{19}}{2*-3}=\frac{44-10\sqrt{19}}{-6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-44)+10\sqrt{19}}{2*-3}=\frac{44+10\sqrt{19}}{-6}
$