2x+3/x-4=6

Solution for 2x+3/x-4=6 equation:

2x+3/x-4=6

We move all terms to the left:

2x+3/x-4-(6)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

2x+3/x-10=0

We multiply all the terms by the denominator


2x*x-10*x+3=0

We add all the numbers together, and all the variables

-10x+2x*x+3=0

Wy multiply elements

2x^2-10x+3=0

a = 2; b = -10; c = +3;
Δ = b2-4ac
Δ = -102-4·2·3
Δ = 76
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:

$\sqrt{\Delta}=\sqrt{76}=\sqrt{4*19}=\sqrt{4}*\sqrt{19}=2\sqrt{19}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-10)-2\sqrt{19}}{2*2}=\frac{10-2\sqrt{19}}{4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-10)+2\sqrt{19}}{2*2}=\frac{10+2\sqrt{19}}{4}
$