If alpha and beta are the roots of x square minus 5x - 1 is equal to zero find the value of Alpha cube + beta cube

Given the quadratic equation:

\[
x^2 - 5x - 1 = 0
\]

Let \(\alpha\) and \(\beta\) be the roots of the equation. We want to find the value of \(\alpha^3 + \beta^3\).

### Step 1: Use Vieta's Formulas
From Vieta's formulas, for the roots of the equation \(ax^2 + bx + c = 0\), we have:

- \(\alpha + \beta = -\frac{b}{a} = 5\)
- \(\alpha \beta = \frac{c}{a} = -1\)

### Step 2: Use the Identity for \(\alpha^3 + \beta^3\)

We use the identity:

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

We need to find \(\alpha^2 + \beta^2\):

\[
\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 5^2 - 2(-1) = 25 + 2 = 27
\]

Now substitute \(\alpha^2 + \beta^2\) into the identity:

\[
\alpha^3 + \beta^3 = ( \alpha + \beta ) \left(\alpha^2 - \alpha \beta + \beta^2\right) = (5) \left(27 - (-1)\right) = 5 \times 28 = 140
\]

Therefore, the value of \(\alpha^3 + \beta^3\) is:

\[
\boxed{140}
\]

Would you like further details or have any other questions?

Here are some related questions that you might find interesting:

1. How do Vieta's formulas apply to cubic equations?
2. Can you find the sum of the squares of the roots of a quadratic equation?
3. What are other identities involving the roots of a polynomial?
4. How can you express higher powers of roots in terms of coefficients?
5. How would the solution change if the quadratic equation had complex roots?

**Tip:** When working with polynomial roots, leveraging identities and relationships like Vieta's formulas can simplify calculations significantly.

Explore how to calculate α^3 + β^3 using Vieta's formulas and cubic identities from the quadratic equation x^2 - 5x - 1 = 0. This step-by-step solution is perfect for students in Grades 9-12, covering key algebraic concepts.

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