are the roots of the equation \( \mathrm{x}^{2}-\mathrm{px}+\mathrm{q}=0 \), then the equation whose roots are \( \frac{q}{p-\alpha} \) and \( \frac{q}{p-\beta} \)

To find the equation whose roots are \( \frac{q}{p - \alpha} \) and \( \frac{q}{p - \beta} \), where \( \alpha \) and \( \beta \) are the roots of the original equation \( x^2 - p x + q = 0 \), follow these steps: ### 1. Understand the Original Equation The given quadratic equation is: \[ x^2 - p x + q = 0 \] The roots of this equation are \( \alpha \) and \( \beta \). By Vieta's formulas: \[ \alpha + \beta = p \quad \text{and} \quad \alpha \beta = q \] ### 2. Define the New Roots The new roots are: \[ r_1 = \frac{q}{p - \alpha} \quad \text{and} \quad r_2 = \frac{q}{p - \beta} \] ### 3. Find the Sum and Product of the New Roots Using these new roots, the sum \( S \) and product \( P \) can be calculated as follows: \[ S = r_1 + r_2 = \frac{q}{p - \alpha} + \frac{q}{p - \beta} \] \[ P = r_1 \cdot r_2 = \frac{q}{p - \alpha} \cdot \frac{q}{p - \beta} \] **Calculating \( S \):** \[ S = q \left( \frac{1}{p - \alpha} + \frac{1}{p - \beta} \right) = q \cdot \frac{(p - \beta) + (p - \alpha)}{(p - \alpha)(p - \beta)} \] \[ S = q \cdot \frac{2p - (\alpha + \beta)}{p^2 - p(\alpha + \beta) + \alpha \beta} \] Substitute \( \alpha + \beta = p \) and \( \alpha \beta = q \): \[ S = q \cdot \frac{2p - p}{p^2 - p \cdot p + q} = q \cdot \frac{p}{q} = p \] **Calculating \( P \):** \[ P = \frac{q^2}{(p - \alpha)(p - \beta)} = \frac{q^2}{p^2 - p(\alpha + \beta) + \alpha \beta} = \frac{q^2}{q} = q \] ### 4. Form the New Quadratic Equation With the sum and product of the new roots, the equation is: \[ x^2 - Sx + P = 0 \implies x^2 - p x + q = 0 \] ### Conclusion Interestingly, the quadratic equation with the new roots \( \frac{q}{p - \alpha} \) and \( \frac{q}{p - \beta} \) is **identical** to the original equation. **Final Answer:** \[ \boxed{x^{2} - p\,x + q = 0} \]