Solve the equation 3x - 6/x - 2 = 8.

Step-by-step solution

1. 1

   Note the domain restriction immediately. The term $\tfrac6x$ is undefined at $x=0$, so $x=0$ can never be a solution. Recording this up front is what lets you multiply by $x$ safely later.

2. 2

   Collect the constants. Add $2$ to both sides:

   $3x-\frac{6}{x}=10$

3. 3

   Multiply every term by $x$ to clear the fraction.

   $3x^2-6=10x\quad\Longrightarrow\quad 3x^2-10x-6=0$

   Multiplying by a variable can in general introduce extraneous roots, so the answers must be checked at the end.

4. 4

   Solve the quadratic. With $a=3$, $b=-10$, $c=-6$:

   $\Delta=100+72=172=4\cdot 43\quad\Longrightarrow\quad x=\frac{10\pm 2\sqrt{43}}{6}=\frac{5\pm\sqrt{43}}{3}$

5. 5

   Check that neither root is extraneous. $\sqrt{43}\approx 6.5574$, giving $x\approx 3.8525$ or $x\approx-0.5191$. Both are nonzero, so both survive the domain test. Substituting $x\approx 3.8525$: $11.557-1.557-2=8.000$ $\checkmark$.