1. If the roots of the equation \( ax^2+bx+c=0 \,(a\ne 0) \) are real and equal, then
(a) \(c=\cfrac{-b}{2a}\) (b) \(c=\cfrac{b}{2a}\) (c) \(c= \cfrac{-b^2}{4a}\) (d) \(c = \cfrac{b^2}{4a}\)
2. 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) -\(\cfrac{1}{3}\)
3. 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
4. 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
5. If the roots of the quadratic equation \(ax^2+bx+c=0\) are real and unequal, the value of \(b^2-4ac\) will be:
(a) >0 (b) <0 (c) 0 (d) None of these
6. 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
7. If the roots of the equation ( ???? + 2 ) ???? 2 − ( ???? − 3 ) ???? + 3 ???? − 1 = 0 are equal in magnitude but opposite in sign, what is the value of ???? ?
(a) -3 (b) 1 (c) 3 (d) None of the above
8. What are the roots of the equation \(x^2 - 4x + 4 = 0\)?
(a) \(2,2\) (b) \(2,-2\) (c) \(\cfrac{1}{2},\cfrac{1}{2}\) (d) \(\cfrac{1}{2},-\cfrac{1}{2}\)
9. If the roots of the equation \(x^2 + 7x + m = 0\) are two consecutive integers, then find the value of \(m\).
10. Translate of your statement in English: If the roots of the equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, then prove that: \(a + c = 2b\).
11. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]
12. Check whether 1 and -1 are roots of the quadratic equation \(x^2 + x + 1 = 0\).
13. Check whether 0 and -2 are roots of the quadratic equation \(8x^2 + 7x = 0\).
14. Check whether \(\cfrac{5}{6}\) and \(\cfrac{4}{3}\) are roots of the equation \(x + \cfrac{1}{x} = \cfrac{13}{6}\).
15. If the sum and product of the roots of the equation \(x^2 - x = k(2x - 1)\) are equal, what is the value of \(k\)?
16. Find the equation whose roots are the squares of the roots of the equation \(x^2 + x + 1 = 0\).
17. If the roots of the quadratic equation \(ax^2 + 2bx + c = 0\) \((a ≠ 0)\) are real and equal, then \(b^2 =\) _____ .
18. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are reciprocals of each other, then \(c =\) _____________.
19. If the roots of the quadratic equation \((1 + m^2)x^2 + 2mcx + (c^2 − a^2) = 0\) are real and equal, show that \(c^2 = a^2(1 + m^2)\).
20. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x + 5 = 0\), then find the value of \((\alpha + \beta)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\right)\).
21. The roots of the equation \(x^2 + x + 1 = 0\) are real.
22. What is the value of \(k\) if the sum and product of the roots of the equation \(kx^2 + 2x + 3k = 0\) \((k \ne 0)\) are equal?
23. Form the equation whose roots are the reciprocals of the roots of the equation \(x^2 + px + 1 = 0\).
24. What should be the value of \(m\) so that the roots of the quadratic equation \(4x^2 + 4(3m - 1)x + (m + 7) = 0\) are reciprocals of each other?
25. If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 - 22x + 105 = 0 \), find the value of \( \alpha - \beta \).
26. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]
27. If the roots of the quadratic equation \(2(a^2 + b^2)x + 2(a + b)x + 1 = 0\) are equal, prove that \(a = b\).
28. 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 \(q\).
29. If the roots of the equation \(ax^2 + bx + 35 = 0\) are -5 and -7, then find the values of \(a\) and \(b\).
30. The roots of the equation \(x^2 - x + 2 = 0\) are not real.
