Find the discriminant of the following quadratic equations, and

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Quadratic Equations

Short Answer Type

71. Find the discriminant of the following quadratic equations, and hence find the nature of its roots.

negative straight x squared plus 2 straight x plus 3 equals 0

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72. Find the discriminant of the following quadratic equations, and hence find the nature of its roots.
$straight x squared plus straight x plus 1 equals 0$

x² + x + 1 = 0
Here, a = 1, b = 1, c = 1
Now, D = b² – 4ac
= (1)² – 4(1) (1) = 1 – 4 = –3
Since, D < 0
Therefore, given quadratic equation has no real roots.

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73. For what value of A, are the roots of the quadratic equation 3x² + 2kx + 27 = 0 real and equal?

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74. For what value of A, are the roots of the quadratic equation: kx² + 4x + 1 = 0 equal and real?

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Long Answer Type

75. For what value of A, does (k – 12)x² + 2 (k – 12)x + 2 = 0 have equal roots?

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Short Answer Type

76. For what value of A, does the equation 9x² + 3kx + 4 = 0 have equal roots?

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Long Answer Type

77. If (–5) is a root of the quadratic equation 2x² + px – 15 = 0 and the quadratic equation p(x² + x) + k = 0 has equal roots, then find the values of p and A.

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78. If one of the roots of the quadratic 2x² + px – 4 = 0 is 4. Write value of p.

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Short Answer Type

79. Find the discriminant of the quadratic equation.

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80. Show that x = – 3 is a solution of x² + 6x + 9 = 0.

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