3x+5/x=12

Solution for 3x+5/x=12 equation:

3x+5/x=12

We move all terms to the left:

3x+5/x-(12)=0


Domain of the equation: x!=0

x∈R


We multiply all the terms by the denominator


3x*x-12*x+5=0

We add all the numbers together, and all the variables

-12x+3x*x+5=0

Wy multiply elements

3x^2-12x+5=0

a = 3; b = -12; c = +5;
Δ = b2-4ac
Δ = -122-4·3·5
Δ = 84
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:

$\sqrt{\Delta}=\sqrt{84}=\sqrt{4*21}=\sqrt{4}*\sqrt{21}=2\sqrt{21}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-12)-2\sqrt{21}}{2*3}=\frac{12-2\sqrt{21}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-12)+2\sqrt{21}}{2*3}=\frac{12+2\sqrt{21}}{6}
$