1. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are real and unequal, then \(b^2 - 4ac\) will be ā
(a) >0 (b) =0 (c) <0 (d) none of the above
2. If the roots of the quadratic equation \(ax^2 + 2bx + c = 0\) \((a ā 0)\) are real and equal, then \(b^2 =\) _____ .
3. What should be the value of \(k\) so that the roots of the quadratic equation \(9x^2 + 3kx + 4 = 0\) are real and equal?
4. Calculate the value(s) of \(k\) for which the quadratic equation \(49x^2 + kx + 1 = 0\) will have real and equal roots.
5. For which value(s) of \(k\) will the quadratic equation \(3x^2 - 5x + 2k = 0\) have real and equal roots?
6. For which value(s) of \(k\) will the quadratic equation \(9x^2 - 24x + k = 0\) have real and equal roots?
7. Calculate the value(s) of \(k\) for which the quadratic equation \(2x^2 + 3x + k = 0\) will have real and equal roots.
8. 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.
9. For which value(s) of \(k\) will the quadratic equation \((3k + 1)x^2 + 2(k + 1)x + k = 0\) have real and equal roots?
10. If the roots of the equation \(x^2 + (p - 3)x + p = 0\) are real and equal, then proveāwithout solvingāthat the value of \(p\) will be either \(1\) or \(9\).
11. If the roots of the quadratic equation \( ax^2 + 2bx - c = 0 \) (where \( a \ne 0 \)) are real and equal, then \( b^2 \) will be āā.
12. The roots of the equation \(ax^2+bx+c=0\) will be real and equal when ā
(a) \(b^2>4ac \) (b) \(b^2=4ac \) (c) \(b^2ā 4ac \) (d) \(b^2<4ac\)
13. For what value of \(k\) will the equation \(2x^2 + 3x + k = 0\) have real and equal roots?
14. If \(b^2 = 4ac\) for the quadratic equation \(ax^2 + bx + c = 0\), then the roots are real and āā.
15. If the roots of the quadratic equation \((1 + m^2)x^2 + 2mcx + (c^2 ā a^2) = 0\) are real and equal, show that \(c^2 = a^2(1 + m^2)\).
16. For which value or values of \(k\) will the following equation have real and equal roots: \[ (3k + 1)x^2 + 2(k + 1)x + k = 0 \]
17. If the roots of the equation \(ax^2 + b + c = 0\) are \(\sin Ī±\) and \(\cos Ī±\), then what is the value of \(b^2\)?
(a) \(a^2-2ac\) (b) \(a^2+2ac\) (c) \(a^2-ac\) (d) \(a^2+ac\)
18. If \(\alpha\) and \(\beta\) are the two roots of the quadratic equation \(3x^2 + 2x - 5 = 0\), then find the value of \(\cfrac{\alpha^2}{\beta} + \cfrac{\beta^2}{\alpha}\).
19. If the roots of the quadratic equation \((1+m^2)x^2 + 2mcx + (c^2 - a^2) = 0\) are real and equal, prove that \(c^2 = a^2(1+m^2)\).
20. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(2x^2 - 3x + 4 = 0\), then what is the value of \(\cfrac{\alpha^2 + \beta^2}{\alpha^{-1} + \beta^{-1}}\)?
21. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(5x^2+2x+3=0\), determine the value of \(\cfrac{\alpha^2}{\beta}+\cfrac{\beta^2}{\alpha}\).
22. 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 \(\cfrac{32}{25}\).
23. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(7x^2 + 5x - 4 = 0\), determine the value of \(\cfrac{\alpha^2}{\beta} + \cfrac{\beta^2}{\alpha}\).
24. In the quadratic equation \(ax^2+bx+c=0 (a \ne 0)\), if \(b^2 = 4ac\), then the roots will be real and _____.
25. If the roots of the equation \(ax^2+bx+c=0\) are \(\alpha\) and \(\beta\), find the value of \(\left(1+\cfrac{\alpha}{\beta}\right)\left(1+\cfrac{\beta}{\alpha}\right)\).
26. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \[ 5x^2 + 2x - 3 = 0, \] find the value of \(\alpha^2 + \beta^2\).
27. If \(5x^2 + 2x - 3 = 0\) is a quadratic equation whose roots are \(\alpha\) and \(\beta\), find the value of \(\alpha^3 + \beta^3\).
28. I will solve the quadratic equation \(5x^2 + 23x + 12 = 0\) using an alternative methodāthat is, by multiplying both sides of the equation by 5 and then finding the roots through the method of completing the square.
29. Write the value of \(k\) for which the quadratic equation \(9x^2 + 3kx + 4 = 0\) has real and equal roots.
30. Determine the value of \(p\) for which the equation \(x^2 + (p - 3)x + p = 0\) will have equal and real roots.
