12+7/x-7+3.x=71

Solution for 12+7/x-7+3.x=71 equation:

12+7/x-7+3.x=71

We move all terms to the left:

12+7/x-7+3.x-(71)=0


Domain of the equation: x!=0

x∈R


We add all the numbers together, and all the variables

3.x+7/x-66=0

We multiply all the terms by the denominator


(3.x)*x-66*x+7=0

We add all the numbers together, and all the variables

(+3.x)*x-66*x+7=0

We add all the numbers together, and all the variables

-66x+(+3.x)*x+7=0

We multiply parentheses

3x^2-66x+7=0

a = 3; b = -66; c = +7;
Δ = b2-4ac
Δ = -662-4·3·7
Δ = 4272
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:

$\sqrt{\Delta}=\sqrt{4272}=\sqrt{16*267}=\sqrt{16}*\sqrt{267}=4\sqrt{267}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-66)-4\sqrt{267}}{2*3}=\frac{66-4\sqrt{267}}{6}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-66)+4\sqrt{267}}{2*3}=\frac{66+4\sqrt{267}}{6}
$