Q27. - Algebra 1

Question

Solve each equation. Check your solutions.

$6x^2=12x$

Step-by-Step Solution

Answer

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

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

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

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 $\mathit{x}\mathbf{=}\mathbf{2}$.

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

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

Step 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 6 x 2 = 12 x .

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

$$\begin{gathered} 6x^2=12x \\ 6x^2-12x=0 \\ 6x\left(x-2\right)=0 \\ x\left(x-2\right)=\frac{0}{6} \\ x\left(x-2\right)=0 \\ x=0\text{ or }x-2=0\left(\text{zero product property}\right) \end{gathered}$$

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 $6x^2=12x$.