Question

Solve each equation. Check your solutions.

$6x^2=12x$

Short answer

The solutions of the given equation $6x^2=12x$are role="math" localid="1647951348338" $\mathit{x}\mathbf{=}\mathbf{0}$and $\mathit{x}\mathbf{=}\mathbf{2}$.

Check the solution $\mathit{x}\mathbf{=}\mathbf{0}$.

role="math" localid="1647951392263" $$\begin{gathered} \mathbf{6}{\mathit{x}}^{\mathbf{2}}\mathbf{=}\mathbf{12}\mathit{x} \\ \mathbf{6}{\mathbf{\left(}\mathbf{0}\mathbf{\right)}}^{\mathbf{2}}\underset{\mathbf{¯}}{\underset{\mathbf{¯}}{\mathbf{?}}}\mathbf{12}\mathbf{\left(}\mathbf{0}\mathbf{\right)} \\ \mathbf{6}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\underset{\mathbf{¯}}{\underset{\mathbf{¯}}{\mathbf{?}}}\mathbf{0} \\ \mathbf{0}\mathbf{=}\mathbf{0} \end{gathered}$$

Therefore, $\mathit{x}\mathbf{=}\mathbf{0}$is the solution of the equation $\mathbf{6}{\mathit{x}}^{\mathbf{2}}\mathbf{=}\mathbf{12}\mathit{x}$.

Check the solution $\mathit{x}\mathbf{=}\mathbf{2}$.

role="math" localid="1647951458750" $$\begin{gathered} \mathbf{}\mathbf{6}{\mathit{x}}^{\mathbf{2}}\mathbf{=}\mathbf{12}\mathit{x} \\ \mathbf{6}{\mathbf{\left(}\mathbf{2}\mathbf{\right)}}^{\mathbf{2}}\underset{\mathbf{¯}}{\underset{\mathbf{¯}}{\mathbf{?}}}\mathbf{12}\mathbf{\left(}\mathbf{2}\mathbf{\right)} \\ \mathbf{6}\mathbf{\left(}\mathbf{4}\mathbf{\right)}\underset{\mathbf{¯}}{\underset{\mathbf{¯}}{\mathbf{?}}}\mathbf{24} \\ \mathbf{\text{24}}\mathbf{=}\mathbf{24} \end{gathered}$$

Therefore,$\mathit{x}\mathbf{=}\mathbf{2}$ is the solution of the equation $\mathbf{6}{\mathit{x}}^{\mathbf{2}}\mathbf{=}\mathbf{12}\mathit{x}$.

Step-by-step solution

Step 1. Write the zero product property.

The zero product property states that if the product of two factors is 0, then at least one of the factors must be 0.

That implies if$ab=0$ then $a=0,b=0$ or both$a$ and$b$ equals to zero.

Step 2. Solve the given equation 6x2=12x.

The solutions of the given equation$6x^2=12x$ are:

role="math" localid="1647951660172" $$\begin{gathered} 6x^2=12x \\ 6x^2-12x=0 \\ 6x\left(x-2\right)=0 \\ x\left(x-2\right)=\frac{0}{6} \\ x\left(x-2\right)=0 \\ x=0\text{or}x-2=0\left(\text{zeroproductproperty}\right) \end{gathered}$$

Therefore, the solutions of the given equation$6x^2=12x$ are $x=0$and $x=2$.

Step 3. Check the solution.

Check the solution $x=0$.

$$\begin{gathered} 6x^2=12x \\ 6{\left(0\right)}^2\underset{¯}{\underset{¯}{?}}12\left(0\right) \\ 6\left(0\right)\underset{¯}{\underset{¯}{?}}0 \\ 0=0 \end{gathered}$$

Therefore, $x=0$is the solution of the equation $6x^2=12x$.

Check the solution $x=2$.

$$\begin{gathered} 6x^2=12x \\ 6{\left(2\right)}^2\underset{¯}{\underset{¯}{?}}12\left(2\right) \\ 6\left(4\right)\underset{¯}{\underset{¯}{?}}24 \\ \text{24}=24 \end{gathered}$$

Therefore,$x=2$ is the solution of the equation $6x^2=12x$.