If α and β are the roots of the equation 2x2 − x − 4 = 0. Find the equation whose roots are α − βα and β − αβ

1. Break Down the Problem

We need to find the roots of the given equation and then determine a new equation based on transformations of these roots.

Given the quadratic equation: $2x^2 - x - 4 = 0$

We know that:

• The roots are $\alpha$ and $\beta$.
• The new roots will be $\alpha - \beta\alpha$ and $\beta - \alpha\beta$.

2. Relevant Concepts

For a quadratic equation $ax^2 + bx + c = 0$, the sum and product of the roots can be given by:

• $\alpha + \beta = -\frac{b}{a}$
• $\alpha\beta = \frac{c}{a}$

From here, we can derive the new roots and form the necessary new equation.

3. Analysis and Detail

First, we need to find $\alpha + \beta$ and $\alpha \beta$:

1. Coefficients from the equation $2x^2 - x - 4 = 0$:

   • $a = 2$
   • $b = -1$
   • $c = -4$

2. Calculate $\alpha + \beta$ and $\alpha \beta$: $\alpha + \beta = -\frac{-1}{2} = \frac{1}{2}$ $\alpha \beta = \frac{-4}{2} = -2$

Next, we will derive the new roots:

• The new roots are given by: $\alpha - \beta \alpha = \alpha(1 - \beta)$ $\beta - \alpha \beta = \beta(1 - \alpha)$

Now we need to calculate the sum and product of these new roots:

1. New root sum: $\alpha(1 - \beta) + \beta(1 - \alpha) = \alpha + \beta - (\alpha\beta + \beta\alpha) = \frac{1}{2} - 2 = \frac{1}{2} - 2 = -\frac{3}{2}$

2. New root product: $\alpha(1 - \beta) \beta(1 - \alpha) = \alpha \beta (1 - \beta)(1 - \alpha)$ Expanding $(1 - \beta)(1 - \alpha)$: $1 - (\alpha + \beta) + \alpha \beta = 1 - \frac{1}{2} - 2 = -\frac{5}{2}$ Thus, $\alpha \beta \cdot (1 - \beta)(1 - \alpha) = -2 \cdot -\frac{5}{2} = 5$

4. Verify and Summarize

We found:

• Sum of new roots = $-\frac{3}{2}$
• Product of new roots = $5$

Using the derived sum and product, we can form the new quadratic equation: $x^2 - \left( -\frac{3}{2} \right)x + 5 = 0$ simplifying, we get: $x^2 + \frac{3}{2}x + 5 = 0$

To eliminate the fraction, we can multiply through by $2$: $2x^2 + 3x + 10 = 0$

Final Answer

The equation whose roots are $\alpha - \beta \alpha$ and $\beta - \alpha \beta$ is: $2x^2 + 3x + 10 = 0$