Solve the equation 3x^2 - x = 0 by factoring.

Step-by-step solution

1. 1

   Notice the missing constant term. Every term contains at least one $x$, which is exactly the condition for pulling $x$ out as a common factor. There is no need for the quadratic formula.

2. 2

   Factor out $x$.

   $3x^2-x=x(3x-1)$

   Check the factoring by expanding back: $x\cdot 3x-x\cdot 1=3x^2-x$ $\checkmark$.

3. 3

   Never divide both sides by $x$. Writing $3x^2=x$ and then cancelling an $x$ gives only $x=\tfrac13$ and throws away the root $x=0$, because dividing by $x$ tacitly assumes $x\neq 0$. Factoring keeps both cases alive.

4. 4

   Apply the zero product property.

   $x(3x-1)=0\ \Longrightarrow\ x=0\ \text{or}\ 3x-1=0$

   $x=0\qquad\text{or}\qquad x=\frac{1}{3}$

5. 5

   Verify both roots.

   $3(0)^2-0=0\quad\checkmark$

   $3\left(\tfrac13\right)^2-\tfrac13=3\cdot\tfrac19-\tfrac13=\tfrac13-\tfrac13=0\quad\checkmark$