3x+6/x-42=0

Solution for 3x+6/x-42=0 equation:

3x+6/x-42=0


Domain of the equation: x!=0

x∈R


We multiply all the terms by the denominator


3x*x-42*x+6=0

We add all the numbers together, and all the variables

-42x+3x*x+6=0

Wy multiply elements

3x^2-42x+6=0

a = 3; b = -42; c = +6;
Δ = b2-4ac
Δ = -422-4·3·6
Δ = 1692
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:

$\sqrt{\Delta}=\sqrt{1692}=\sqrt{36*47}=\sqrt{36}*\sqrt{47}=6\sqrt{47}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-42)-6\sqrt{47}}{2*3}=\frac{42-6\sqrt{47}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-42)+6\sqrt{47}}{2*3}=\frac{42+6\sqrt{47}}{6}
$