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To solve this problem, we need to use Vieta's formulas and the identity for the sum of the cubes of the roots of a cubic equation.

Given the cubic equation:

\[
2x^3 + 4x - 3 = 0
\]

Let the roots of this equation be \( \alpha \), \( \beta \), and \( \gamma \).

### Step 1: Vieta's Formulas
From Vieta's formulas for a cubic equation \( ax^3 + bx^2 + cx + d = 0 \), we have:
- \( \alpha + \beta + \gamma = -\frac{b}{a} \)
- \( \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \)
- \( \alpha\beta\gamma = -\frac{d}{a} \)

In our case, the coefficients are:
- \( a = 2 \)
- \( b = 0 \) (since there is no \(x^2\) term)
- \( c = 4 \)
- \( d = -3 \)

Using Vieta's formulas:
1. \( \alpha + \beta + \gamma = -\frac{0}{2} = 0 \)
2. \( \alpha\beta + \beta\gamma + \gamma\alpha = \frac{4}{2} = 2 \)
3. \( \alpha\beta\gamma = -\frac{-3}{2} = \frac{3}{2} \)

### Step 2: Calculate \( \alpha^3 + \beta^3 + \gamma^3 \)
We use the identity for the sum of cubes of the roots:

\[
\alpha^3 + \beta^3 + \gamma^3 = (\alpha + \beta + \gamma)(\alpha^2 + \beta^2 + \gamma^2 - \alpha\beta - \beta\gamma - \gamma\alpha) + 3\alpha\beta\gamma
\]

Given \( \alpha + \beta + \gamma = 0 \), the equation simplifies to:

\[
\alpha^3 + \beta^3 + \gamma^3 = 3\alpha\beta\gamma
\]

Substituting the known value of \( \alpha\beta\gamma \):

\[
\alpha^3 + \beta^3 + \gamma^3 = 3 \times \frac{3}{2} = \frac{9}{2}
\]

Thus, the value of \( \alpha^3 + \beta^3 + \gamma^3 \) is \( \boxed{\frac{9}{2}} \).

Would you like more details or have any questions?

Here are some related questions for further practice:

1. How can you find the sum of squares of the roots of a cubic equation using Vieta's formulas?
2. What is the role of the constant term in a cubic equation in determining the product of the roots?
3. How does the absence of an \(x^2\) term affect the sum of the roots in a cubic equation?
4. Can the sum of cubes of the roots be negative? If so, under what conditions?
5. How would the identity for the sum of cubes change if the cubic equation had a different form?

**Tip:** Always simplify identities using known values like the sum or product of roots to make calculations easier!

If α, β, and γ are the roots of the cubic equation 2x^3 + 4x - 3 = 0, find the value of α^3 + β^3 + γ^3.

Learn how to calculate the sum of cubes of the roots for the cubic equation 2x^3 + 4x - 3 = 0 using Vieta's formulas and other algebraic identities. Ideal for high school students studying cubic equations and their properties.

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