3x-19/x-5=x-3/3

Solution for 3x-19/x-5=x-3/3 equation:

3x-19/x-5=x-3/3

We move all terms to the left:

3x-19/x-5-(x-3/3)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

3x-19/x-(x-1)-5=0

We get rid of parentheses

3x-19/x-x+1-5=0

We multiply all the terms by the denominator


3x*x-x*x+1*x-5*x-19=0

We add all the numbers together, and all the variables

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

Wy multiply elements

3x^2-1x^2-4x-19=0

We add all the numbers together, and all the variables

2x^2-4x-19=0

a = 2; b = -4; c = -19;
Δ = b2-4ac
Δ = -42-4·2·(-19)
Δ = 168
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:

$\sqrt{\Delta}=\sqrt{168}=\sqrt{4*42}=\sqrt{4}*\sqrt{42}=2\sqrt{42}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-4)-2\sqrt{42}}{2*2}=\frac{4-2\sqrt{42}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-4)+2\sqrt{42}}{2*2}=\frac{4+2\sqrt{42}}{4}
$