We are given that $\alpha$ and $\beta$ are the roots (zeroes) of the quadratic polynomial $x^2 - x - 4$. From Vieta's formulas, we know the following relationships for a quadratic equation $ax^2 + bx + c = 0$:

• Sum of the roots: $\alpha + \beta = -\frac{b}{a}$
• Product of the roots: $\alpha\beta = \frac{c}{a}$

For the quadratic equation $x^2 - x - 4$, we have:

• $a = 1$, $b = -1$, and $c = -4$.

From Vieta’s formulas: $\alpha + \beta = -\frac{-1}{1} = 1$ $\alpha\beta = \frac{-4}{1} = -4$

We are asked to find the value of $\frac{1}{\alpha} + \frac{1}{\beta} - \alpha\beta$.

Step 1: Simplify the Expression

We can combine the terms $\frac{1}{\alpha} + \frac{1}{\beta}$ by using a common denominator: $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta}$

Thus, the given expression becomes: $\frac{1}{\alpha} + \frac{1}{\beta} - \alpha\beta = \frac{\alpha + \beta}{\alpha \beta} - \alpha\beta$

Step 2: Substitute the Known Values

We already know:

• $\alpha + \beta = 1$
• $\alpha \beta = -4$

Substitute these values into the expression: $\frac{1}{\alpha} + \frac{1}{\beta} - \alpha\beta = \frac{1}{-4} - (-4)$

This simplifies to: $\frac{1}{-4} + 4 = -\frac{1}{4} + 4 = -\frac{1}{4} + \frac{16}{4} = \frac{15}{4}$

Thus, the value of $\frac{1}{\alpha} + \frac{1}{\beta} - \alpha\beta$ is $\frac{15}{4}$.

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Would you like a more detailed explanation or have any questions? Here are some follow-up questions you may find useful:

1. How do Vieta’s formulas relate to solving quadratic equations?
2. What is the role of the discriminant in determining the nature of the roots of a quadratic equation?
3. How can we apply this to cubic or higher-degree polynomials?
4. What are other methods to solve a quadratic equation, apart from using Vieta’s formulas?
5. How do you find the roots of a quadratic equation when the sum and product of the roots are known?

Tip: Always remember that for any quadratic equation $ax^2 + bx + c = 0$, the sum and product of the roots are easy to find using Vieta's formulas: $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$.