If \( \alpha, \beta \) are the roots of the equation \( 2 x^{2}+6 x-10=0 \). Find \( \alpha^{4}+\beta^{4} \) (b) \( \alpha^{3}+\beta^{3} \) (c) \( \alpha^{6}+\beta^{6} \) (d) \( \alpha^{6}-\beta^{6} \)

To solve the problem, we can use the relationships between the roots of a quadratic equation and Vieta's formulas. Given the equation \( 2x^2 + 6x - 10 = 0 \), we first find the roots \( \alpha \) and \( \beta \). 1. **Finding \( \alpha + \beta \) and \( \alpha \beta \)**: By Vieta's formulas, we have: \[ \alpha + \beta = -\frac{b}{a} = -\frac{6}{2} = -3, \] \[ \alpha \beta = \frac{c}{a} = \frac{-10}{2} = -5. \] 2. **Finding \( \alpha^2 + \beta^2 \)**: We can use the identity: \[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta = (-3)^2 - 2(-5) = 9 + 10 = 19. \] 3. **Finding \( \alpha^3 + \beta^3 \)**: We use the identity for the sum of cubes: \[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 + \beta^2 - \alpha \beta). \] So, substituting: \[ \alpha^3 + \beta^3 = (-3)(19 - (-5)) = -3(19 + 5) = -3 \times 24 = -72. \] 4. **Finding \( \alpha^4 + \beta^4 \)**: We use the relation: \[ \alpha^4 + \beta^4 = (\alpha^2 + \beta^2)^2 - 2(\alpha \beta)^2. \] We already found \( \alpha^2 + \beta^2 = 19 \) and \( \alpha \beta = -5 \): \[ \alpha^4 + \beta^4 = 19^2 - 2(-5)^2 = 361 - 50 = 311. \] 5. **Finding \( \alpha^6 + \beta^6 \)**: Using the recurrence relation \( \alpha^n + \beta^n = (\alpha + \beta)(\alpha^{n-1} + \beta^{n-1}) - (\alpha \beta)(\alpha^{n-2} + \beta^{n-2}) \): \[ S_n = \alpha^n + \beta^n. \] We know: - \( S_0 = 2 \), - \( S_1 = -3 \), - \( S_2 = 19 \), - \( S_3 = -72 \). Now, calculating \( S_4 \): \[ S_4 = (-3)(-72) - (-5)(19) = 216 + 95 = 311. \] Now, calculating \( S_5 \): \[ S_5 = (-3)(311) - (-5)(-72) = -933 - 360 = -1293. \] Finally, calculating \( S_6 \): \[ S_6 = (-3)(-1293) - (-5)(311) = 3879 + 1555 = 5434. \] 6. **Finding \( \alpha^6 - \beta^6 \)**: For this difference, we can use the formula: \[ \alpha^6 - \beta^6 = (\alpha - \beta)(\alpha^5 + \beta^5) \] First, we need \( \alpha^5 + \beta^5 \): \[ S_5 = -1293, \text{ and } (\alpha - \beta) = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} = \sqrt{(-3)^2 - 4(-5)} = \sqrt{9 + 20} = \sqrt{29}. \] So, finally: \[ \alpha^6 - \beta^6 = \sqrt{29} \times S_5 = \sqrt{29} \times (-1293). \] To summarize the results, we have: - (a) \( \alpha^4 + \beta^4 = 311 \) - (b) \( \alpha^3 + \beta^3 = -72 \) -