Question

Solve:

$8x^3=24x^2-18x$

Short answer

The solutions of the quadratic equations $8x^3=24x^2-18x$are $x=0,\frac{3}{2}$

Step-by-step solution

Step 1. Given Information.

An equation:

$8x^3=24x^2-18x$

Step 2. Write the quadratic equation in the standard form.

$$\begin{gathered} 8x^3=24x^2-18x \\ 8x^3-24x^2+18x=0 \\ 2x(4x^2-12x+9)=0 \end{gathered}$$

Step 3. Factor the quadratic expression.

$$\begin{gathered} 2x(4x^2-12x+9)=0 \\ 2x(4x^2-6x-6x+9)=0 \\ 2x{(2x-3)}^2=0 \end{gathered}$$

Step 4. Use the Zero-Product property.

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

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

The value ofx is $x=0,\frac{3}{2}$

Step 5. Check the answer.

Check by substituting the values in the original equation,

When $x=0$,

$$\begin{gathered} 8\left(0\right)=24\left(0\right)-18\left(0\right) \\ 0=0 \end{gathered}$$

When $x=\frac{3}{2}$,

$$\begin{gathered} 8{\left(\frac{3}{2}\right)}^3=24{\left(\frac{3}{2}\right)}^2-18\left(\frac{3}{2}\right) \\ 27=54-27 \end{gathered}$$

Hence the equations hold true.