Given that $\alpha$ and $\beta$ are the roots of the equation $x^2 - x - 3 = 0$, we know that:

$\alpha + \beta = 1 \quad \text{(sum of the roots)}$ $\alpha \beta = -3 \quad \text{(product of the roots)}$

We are asked to find the equation with roots $\frac{\alpha}{\beta^2}$ and $\frac{\beta}{\alpha^2}$.

Step 1: Sum of the new roots

The sum of the new roots is:

= \frac{\alpha^3 + \beta^3}{\alpha^2 \beta^2}$$ Using the identity for the sum of cubes: $$\alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2)$$ We already know $$\alpha + \beta = 1$$, so: $$\alpha^3 + \beta^3 = (1)(\alpha^2 - \alpha\beta + \beta^2)$$ Also: $$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 1^2 - 2(-3) = 1 + 6 = 7$$ $$\alpha^3 + \beta^3 = 7 - (-3) = 10$$ Thus: $$\frac{\alpha}{\beta^2} + \frac{\beta}{\alpha^2} = \frac{10}{\alpha^2\beta^2} = \frac{10}{9}$$ ### Step 2: Product of the new roots The product of the new roots is: $$\frac{\alpha}{\beta^2} \cdot \frac{\beta}{\alpha^2} = \frac{\alpha \beta}{\alpha^2 \beta^2} = \frac{1}{\alpha \beta} = \frac{1}{-3} = -\frac{1}{9}$$ ### Step 3: Forming the quadratic equation The quadratic equation with roots $$\frac{\alpha}{\beta^2}$$ and $$\frac{\beta}{\alpha^2}$$ is given by: $$x^2 - \left(\text{sum of roots}\right) x + \left(\text{product of roots}\right) = 0$$ Substituting the values: $$x^2 - \frac{10}{9}x - \left(-\frac{1}{9}\right) = 0$$ Multiplying the entire equation by 9 to eliminate the fractions: $$9x^2 - 10x + 1 = 0$$ So, the required equation is: $$9x^2 - 10x + 1 = 0$$ Would you like further details on this solution, or do you have any questions? Here are five related questions for further exploration: 1. How do we derive the identity for the sum of cubes? 2. What is the significance of the discriminant in determining the nature of roots? 3. How can we generalize this problem to higher-degree polynomials? 4. What is the geometric interpretation of the roots of the equation? 5. How would the equation change if the original roots were complex numbers? **Tip:** When working with roots of quadratic equations, always remember to verify the relationships between the sum and product of the roots, as they are key in forming new equations.