3x-19/x-3=x-4/4

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

3x-19/x-3=x-4/4

We move all terms to the left:

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


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

-2x+3x*x-x*x-19=0

Wy multiply elements

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

We add all the numbers together, and all the variables

2x^2-2x-19=0

a = 2; b = -2; c = -19;
Δ = b2-4ac
Δ = -22-4·2·(-19)
Δ = 156
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:

$\sqrt{\Delta}=\sqrt{156}=\sqrt{4*39}=\sqrt{4}*\sqrt{39}=2\sqrt{39}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-2)-2\sqrt{39}}{2*2}=\frac{2-2\sqrt{39}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-2)+2\sqrt{39}}{2*2}=\frac{2+2\sqrt{39}}{4}
$