19+3/5x=x-3

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

19+3/5x=x-3

We move all terms to the left:

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


Domain of the equation: 5x!=0

x!=0/5

x!=0

x∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


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

Wy multiply elements

-5x^2+15x+95x+3=0

We add all the numbers together, and all the variables

-5x^2+110x+3=0

a = -5; b = 110; c = +3;
Δ = b2-4ac
Δ = 1102-4·(-5)·3
Δ = 12160
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:

$\sqrt{\Delta}=\sqrt{12160}=\sqrt{64*190}=\sqrt{64}*\sqrt{190}=8\sqrt{190}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(110)-8\sqrt{190}}{2*-5}=\frac{-110-8\sqrt{190}}{-10}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(110)+8\sqrt{190}}{2*-5}=\frac{-110+8\sqrt{190}}{-10}
$