1. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x + 5 = 0\), then find the value of \((\alpha + \beta)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\right)\).

2. If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 - 22x + 105 = 0 \), find the value of \( \alpha - \beta \).

3. If \(\alpha\) and \(\beta\) are the two roots of the quadratic equation \(3x^2 + 2x - 5 = 0\), then find the value of \(\cfrac{\alpha^2}{\beta} + \cfrac{\beta^2}{\alpha}\).

4. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x - 10 = 0\), then find \(\alpha^2 + \beta^2\).

5. If \(α\) and \(β\) are the roots of the equation \(x^2 - 22x + 105 = 0\), find the value of \((α - β)\).

6. If \( \alpha \) and \( \beta \) are the roots of the equation \( 5x^2 - 3x + 6 = 0 \), Find the value of \( \left( \frac{1}{\alpha} + \frac{1}{\beta} \right) \).

7. For the equation \( 3x^2 + 8x + 2 = 0 \), if the roots are \( \alpha \) and \( \beta \), then what is the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \)?"

(a) -\(\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

8. If \(\alpha\) and \(\beta\) are the roots of the equation \(5x^2 + 2x - 3 = 0\), then the value of \(\alpha^2 + \beta^2\) will be \(\cfrac{32}{25}\).

9. If \(\alpha\) and \(\beta\) are the roots of the equation \(3x^2+8x+2=0\), find the value of \(\cfrac{1}{\alpha^2}+\cfrac{1}{\beta^2}\).

10. If the quadratic equation \(x^2 + px + q = 0\) has roots \(\alpha\) and \(\beta\), then find the value of \(\alpha^3 + \beta^3\).

11. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation with roots \(\alpha\) and \(\beta\), find the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\).

12. If one root of the equation \(x^2 + px + 12 = 0\) is \(2\), and both roots of the equation \(x^2 + px + q = 0\) are equal, then find the value of \(q\).

13. If \(\alpha\) and \(\beta\) are the roots of the equation \(5x^2-3x+6=0\), determine the value of \(\left(\cfrac{1}{\alpha}+\cfrac{1}{\beta}\right)\).

14. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2x^2 - 3x + 4 = 0\), then what is the value of \(\cfrac{\alpha^2 + \beta^2}{\alpha^{-1} + \beta^{-1}}\)?

15. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(5x^2+2x+3=0\), determine the value of \(\cfrac{\alpha^2}{\beta}+\cfrac{\beta^2}{\alpha}\).

16. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(7x^2 + 5x - 4 = 0\), determine the value of \(\cfrac{\alpha^2}{\beta} + \cfrac{\beta^2}{\alpha}\).

17. If the roots of the equation \(ax^2+bx+c=0\) are \(\alpha\) and \(\beta\), find the value of \(\left(1+\cfrac{\alpha}{\beta}\right)\left(1+\cfrac{\beta}{\alpha}\right)\).

18. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \[ 5x^2 + 2x - 3 = 0, \] find the value of \(\alpha^2 + \beta^2\).

19. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation whose roots are \(\alpha\) and \(\beta\), find the value of \(\alpha^3 + \beta^3\).

20. If \( \alpha \) and \( \beta \) are the roots of the equation \( ax^2 + bx + c = 0 \), then find the value of \[ \frac{a\alpha^2}{b\alpha + c} - \frac{a\beta^2}{b\beta + c} \]

21. Got it — sticking strictly to translation. Here's the English version without any extra commentary: If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( 5x^2 - 3x + 6 = 0 \), then \( \alpha + \beta = -\frac{-3}{5} = \frac{3}{5} \) and \( \alpha\beta = \frac{6}{5} \) \(\therefore \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha\beta} \) \( = \frac{\frac{3}{5}}{\frac{6}{5}} = \frac{3}{5} \times \frac{5}{6} = \frac{1}{2} \) (Answer)

22. If the roots of the equation \(x^2 + ax + b = 0\) are \(-2\) and \(3\), then find the values of \(a\) and \(b\).

(a) 1,6 (b) -1,-6 (c) -1,6 (d) 1,-6

23. If \(α\) and \(β\) are the roots of the equation \(3x^2 + 8x + 2 = 0\), find the value of \(\cfrac{1}{α} + \cfrac{1}{β}\).

(a) \(-\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

24. If the roots of the equation \(x^2 + 7x + m = 0\) are two consecutive integers, then find the value of \(m\).

25. If the roots of the equation \(ax^2 + bx + 35 = 0\) are -5 and -7, then find the values of \(a\) and \(b\).

26. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+7x+3=0\), prove that: \[ \alpha^3+\beta^3+7(\alpha^2+\beta^2)+3(\alpha+\beta)=0 \]

27. If the roots of the equation \(3x^2+8x+2=0\) are \(\alpha\) and \(\beta\), then determine the equation whose roots are \(\cfrac{1}{\alpha}\) and \(\cfrac{1}{\beta}\).

28. If \(\alpha\) and \(\beta\) are the roots of the equation \(3x^2 - 8x + 2 = 0\), determine the equation whose roots are \(\cfrac{\alpha}{\beta}\) and \(\cfrac{\beta}{\alpha}\).

29. If \(\alpha\) and \(\beta\) are the two roots of the equation \(ax^2 + bx + c = 0\), then \(\cfrac{\alpha}{\beta}\) and \(\cfrac{\beta}{\alpha}\) are also roots of a quadratic equation. Determine that quadratic equation.

30. If \(\alpha, \beta\) are the roots of the equation \(3x^2+8x+2=0\), then the value of \( \cfrac{1}{\alpha}+\cfrac{1}{\beta}\) is –

(a) \(-\cfrac{3}{8}\) (b) 4 (c) \(\cfrac{2}{3}\) (d) -4