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The equation provided is:

\[
4x^{\frac{7}{3}} + 3x^{\frac{4}{3}} - x^{\frac{1}{3}} = 0
\]

To solve this equation, we can approach it by factoring. First, observe that each term has a common factor of \(x^{\frac{1}{3}}\). We can factor out \(x^{\frac{1}{3}}\) from the equation:

\[
x^{\frac{1}{3}} \left( 4x^{2} + 3x - 1 \right) = 0
\]

This gives us two possible cases:

1. \(x^{\frac{1}{3}} = 0\)
2. \(4x^2 + 3x - 1 = 0\)

### Case 1: \(x^{\frac{1}{3}} = 0\)

This implies:

\[
x = 0
\]

### Case 2: \(4x^2 + 3x - 1 = 0\)

This is a quadratic equation, which we can solve using the quadratic formula:

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

For the equation \(4x^2 + 3x - 1 = 0\), the coefficients are \(a = 4\), \(b = 3\), and \(c = -1\). Substituting these values into the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{3^2 - 4(4)(-1)}}{2(4)}
\]

\[
x = \frac{-3 \pm \sqrt{9 + 16}}{8}
\]

\[
x = \frac{-3 \pm \sqrt{25}}{8}
\]

\[
x = \frac{-3 \pm 5}{8}
\]

So, we have two solutions:

\[
x = \frac{-3 + 5}{8} = \frac{2}{8} = \frac{1}{4}
\]

\[
x = \frac{-3 - 5}{8} = \frac{-8}{8} = -1
\]

### Final Solutions

The solutions to the equation are:

\[
x = 0, \quad x = \frac{1}{4}, \quad \text{and} \quad x = -1
\]

Would you like more details on any part of the solution, or do you have any other questions?

### Related Questions

1. How do you factor polynomial expressions like the one in this problem?
2. What are the general steps for solving quadratic equations?
3. Can you explain why we factored out \(x^{\frac{1}{3}}\) in this equation?
4. What are some other methods to solve cubic equations?
5. How can the quadratic formula be derived?
6. What is the importance of checking for common factors in equations?
7. Are there special cases where the quadratic formula does not apply?
8. How does the discriminant in the quadratic formula affect the number of real solutions?

### Tip
Always look for common factors before attempting to solve equations—it can simplify the problem significantly.

Learn how to solve the equation 4x^(7/3) + 3x^(4/3) - x^(1/3) = 0 step-by-step with detailed solutions. This problem involves factoring polynomials and solving quadratic equations. Suitable for high school students.

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