Question

Solve:

$6a^2+9a=3a$

Short answer

The solutions of the quadratic equations $6a^2+9a=3a$are $a=0,-1$

Step-by-step solution

Step 1. Given Information

An equation:

$6a^2+9a=3a$

Step 2. Write the quadratic equation in standard form.

$$\begin{gathered} 6a^2+9a=3a \\ 6a^2+9a-3a=0 \\ 6a^2+6a=0 \end{gathered}$$

Step 3. Factor the quadratic expression

Factorize the equation,

$$\begin{gathered} 6a^2+6a=0 \\ 6a(a+1)=0 \end{gathered}$$

Step 4. Use the Zero-Product property

Use the Zero Product Property and set each factor to zero.

$$\begin{gathered} 6a=0 \\ a=0 \\ a+1=0 \\ a=-1 \end{gathered}$$

The value of a is $a=0,-1$

Step 5. Check the answer

Check by substituting the values in the original equation,

When $a=-1,$

$$\begin{gathered} 6{(-1)}^2+9(-1)=3(-1) \\ 6-9=-3 \\ -3=-3 \end{gathered}$$

When $a=0$,

$6{\left(0\right)}^2+9{\left(0\right)}^2=3{\left(0\right)}^{2}0=0$

Hence the equations hold true.