Q. 297 - Intermediate Algebra

Question

In the following exercises, solve:

$\left(2x+1\right)\left(x-3\right)=-4x$

Step-by-Step Solution

Answer

On solving the equation, we get $x=-1,\frac{3}{2}$.

Step 1. Given information.

Consider the given equation,

$\left(2x+1\right)\left(x-3\right)=-4x$

Write the quadratic equation in standard form,

$$\begin{gathered} 2x^2-6x+x-3=-4x \\ 2x^2-9x-3=0 \end{gathered}$$

Step 2. Factor the quadratic expression.

On factoring, we get,

$$\begin{gathered} 2x^2-3x+2x-3=0 \\ x\left(2x-3\right)+\left(2x-3\right)=0 \\ \left(x+1\right)\left(2x-3\right)=0 \end{gathered}$$

Use the zero product property,

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

Then,

$$\begin{gathered} 2x-3=0 \\ x=\frac{3}{2} \end{gathered}$$

Step 3. Check the answers.

Substitute $x=-1$ in the given equation,

$$\begin{gathered} \left(2\left(-1\right)+1\right)\left(-1-3\right)=-4\left(-1\right) \\ \left(-1\right)\left(-4\right)=4 \\ 4=4 \end{gathered}$$

This is true.

Then,

$$\begin{gathered} \left(2\left(\frac{3}{2}\right)+1\right)\left(\frac{3}{2}-3\right)=-4\left(\frac{3}{2}\right) \\ \left(3+1\right)\left(-\frac{3}{2}\right)=-6 \\ -6=-6 \end{gathered}$$

This is also true.

Hence, the answer is $x=-1,\frac{3}{2}$.