1. Let us check whether \(1\) and \(-1\) are roots of the quadratic equation \(x^2 + x + 1 = 0\).

2. Check whether \(\cfrac{5}{6}\) and \(\cfrac{4}{3}\) are roots of the equation \(x + \cfrac{1}{x} = \cfrac{13}{6}\).

3. Find the equation whose roots are the squares of the roots of the equation \(x^2 + x + 1 = 0\).

4. If the roots of the quadratic equation \((1 + m^2)x^2 + 2mcx + (c^2 − a^2) = 0\) are real and equal, show that \(c^2 = a^2(1 + m^2)\).

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

6. Form the equation whose roots are the reciprocals of the roots of the equation \(x^2 + px + 1 = 0\).

7. 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 \(q\).

8. Form the equation whose roots are the reciprocals of the roots of the equation \(x^2 + mx + 1 = 0\).

9. If the roots of the quadratic equation \((1+m^2)x^2 + 2mcx + (c^2 - a^2) = 0\) are real and equal, prove that \(c^2 = a^2(1+m^2)\).

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

11. Determine the equation whose roots are the reciprocals of the roots of the equation \(x^2 + px + 1 = 0\).

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

13. Calculate the value(s) of \(k\) for which the quadratic equation \(x^2 - 2(5 + 2k)x + 3(7 + 10k) = 0\) will have real and equal roots.

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

15. Express the equation \(x^3 - 4x^2 - x + 1 = (x + 2)^3\) in the standard form of a quadratic equation and write the coefficients of \(x^2\), \(x\), and \(x^0\).

16. I will solve the quadratic equation \(5x^2 + 23x + 12 = 0\) using an alternative method—that is, by multiplying both sides of the equation by 5 and then finding the roots through the method of completing the square.

17. If \( \alpha \) and \( \beta \) are the roots of the equation \(x^2 - 22x + 105 = 0\), then find the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \).

18. Prove that:The squares of the roots of the equation \(x^2 + x + 1 = 0\) are also roots of the same equation \(x^2 + x + 1 = 0\). Would you like me to walk through the proof again in English, or are you asking for a translated version of your original Bengali explanation? I’ve got both options ready.

19. If the roots of the equation \(x^2 + (p - 3)x + p = 0\) are real and equal, then prove—without solving—that the value of \(p\) will be either \(1\) or \(9\).

20. 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)

21. For the equation \(5x^2+9x+3=0\) , if the roots are \(α\) and \(β\), then what is the value of \(\cfrac{1}{α}+\cfrac{1}{β}\) ?

(a) 3 (b) -3 (c) \(\cfrac{1}{3}\) (d) -\(\cfrac{1}{3}\)

22. 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

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. The roots of the equation \(x^2 - 18x + 8 = 0\) are —

(a) Real , Rational , Unequal (b) equal,Rational (c) Real , Rational , equal (d) None of the above

25. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]

26. Write whether the equation \(x - 1 + \frac{1}{x} = 6,\ (x ≠ 0)\) can be expressed in the form \(ax^2 + bx + c = 0\), where \(a, b, c\) are real numbers and \(a ≠ 0\).

27. Translate: Express \((x + 2)^3 = x(x^2 - 1)\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\), and write down the coefficients of \(x^2\), \(x\), and \(x^0\) (i.e., the constant term).

28. Check whether 0 and -2 are roots of the quadratic equation \(8x^2 + 7x = 0\).

29. If \(5x^2 − 2x + 3 = 0\) is a quadratic equation with roots \(α\) and \(β\), find the value of \(\frac{1}{α} + \frac{1}{β}\).

30. The roots of the equation \(x^2 + x + 1 = 0\) are real.