To solve the equation \( \frac{4}{x} - \frac{5}{3} = \frac{x}{6} \), follow these steps:

1. **Eliminate the fractions** by finding a common denominator. The denominators here are \(x\), \(3\), and \(6\). The least common multiple (LCM) of these denominators is \(6x\).

2. **Multiply every term** in the equation by \(6x\) to clear the fractions:

\[
6x \cdot \frac{4}{x} - 6x \cdot \frac{5}{3} = 6x \cdot \frac{x}{6}
\]

Simplifying each term:

\[
6 \cdot 4 - 2x \cdot 5 = x^2
\]

This simplifies to:

\[
24 - 10x = x^2
\]

3. **Rearrange the equation** into a standard quadratic form:

\[
x^2 + 10x - 24 = 0
\]

4. **Solve the quadratic equation** using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 1\), \(b = 10\), and \(c = -24\). Plug these values into the quadratic formula:

\[
x = \frac{-10 \pm \sqrt{10^2 + 4 \cdot 1 \cdot 24}}{2 \cdot 1}
\]
\[
x = \frac{-10 \pm \sqrt{100 + 96}}{2}
\]
\[
x = \frac{-10 \pm \sqrt{196}}{2}
\]
\[
x = \frac{-10 \pm 14}{2}
\]

This gives us two solutions:

\[
x = \frac{-10 + 14}{2} = 2
\]

\[
x = \frac{-10 - 14}{2} = -12
\]

5. **Check the solutions** by substituting them back into the original equation to ensure they work.

For \(x = 2\):

\[
\frac{4}{2} - \frac{5}{3} = \frac{2}{6}
\]

\(\frac{2}{1.5}\) is not equal to \(\frac{2}{6}\), so this is not a solution.

For \(x = -12\):

\[
\frac{4}{-12} - \frac{5}{3} = \frac{-12}{6}
\]

\(\frac{-1}{3}\) is equal to \(\frac{-12}{6}\), so this is a solution.

Therefore, the solution is \(x = -12\).

Would you like more details or have any other questions?

Here are 5 related questions:
1. How do you factor a quadratic equation?
2. What are some alternative methods for solving quadratic equations?
3. How does the quadratic formula derive from completing the square?
4. What is the importance of checking potential solutions in an equation?
5. How does multiplying by the LCM help solve equations with fractions?

**Tip:** Always check potential solutions by substituting them back into the original equation to avoid extraneous solutions.

Learn how to solve the equation 4/x - 5/3 = x/6 with detailed step-by-step instructions. This problem involves fractions and quadratic equations, and is suitable for advanced high school students. Check the solution and understand the process.

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