Q. 299 - Intermediate Algebra

Question

In the following exercises, solve:

$\left(3x-2\right)\left(x+4\right)=12x$

Step-by-Step Solution

Answer

On solving the equation, we get $x=2,-\frac{4}{3}$.

Step 1. Given information.

Consider the given equation,

$\left(3x-2\right)\left(x+4\right)=12x$

Write the quadratic equation in standard form,

$$\begin{gathered} 3x^2+12x-2x-8-12x=0 \\ 3x^2-2x-8=0 \end{gathered}$$

Step 2. Factor the quadratic expression.

On factoring, we get,

$$\begin{gathered} 3x^2-6x+4x-8=0 \\ 3x\left(x-2\right)+4\left(x-2\right)=0 \\ \left(3x+4\right)\left(x-2\right)=0 \end{gathered}$$

Use the zero product property,

$$\begin{gathered} 3x+4=0 \\ x=-\frac{4}{3} \end{gathered}$$

Then,

$$\begin{gathered} x-2=0 \\ x=2 \end{gathered}$$

Step 3. Check the answers.

Substitute $x=-\frac{4}{3}$ the given equation,

$$\begin{gathered} \left(3\left(-\frac{4}{3}\right)-2\right)\left(-\frac{4}{3}+4\right)=12\left(-\frac{4}{3}\right) \\ \left(-6\right)\left(\frac{8}{3}\right)=-16 \\ -16=-16 \end{gathered}$$

This is true.

Substitute $x=2$ in the given equation,

$$\begin{gathered} \left(3\left(2\right)-2\right)\left(2+4\right)=12\left(2\right) \\ \left(4\right)\left(6\right)=12\left(2\right) \\ 24=24 \end{gathered}$$

This is also true.

Hence, the answer is $x=2,-\frac{4}{3}$.