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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

109 Views

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

92 Views

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

547 Views

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

224 Views

Long Answer Type

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

375 Views

Short Answer Type

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

626 Views

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.

117 Views

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

Let the roots of the quadratic equation be α and β.
We have,
2x² + px – 4 = 0, then
Product of the roots = $fraction numerator negative 4 over denominator 2 end fraction equals negative 2$
$rightwards double arrow$ $straight alpha. space straight beta equals negative 2$
But $straight alpha equals 4$
$rightwards double arrow space space space space 4 space left parenthesis straight beta right parenthesis space equals space minus 2$
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Sum of the roots = $fraction numerator negative straight p over denominator 2 end fraction$
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$space rightwards double arrow space space space space space space space space space space space space space 4 minus 1 half equals fraction numerator negative straight p over denominator 2 end fraction space rightwards double arrow space space space space space space space space space space space space space space space space fraction numerator negative straight p over denominator 2 end fraction space equals space 7 over 2 space rightwards double arrow space space space space space space space space space space space space space space space space space space space space straight p space equals space minus 7$