Q. 13

# If – 4 is a root of the equation x2 + 2x + 4p = 0, find the value of k for which the quadratic equation x2 + px (1 + 3k) + 7(3 + 2k) = 0 has equal roots.

Given – 4 is a root of the equation x2 + 2x + 4p = 0

(– 4)2 + 2(– 4) + 4p = 0

8 + 4p = 0

p = – 2

The quadratic equation x2 + px (1 + 3k) + 7(3 + 2k) = 0 has equal roots

Comparing with standard quadratic equation ax2 + bx + c = 0

a = 1 b = p(1 + 3k) c = 7(3 + 2k)

Thus D = 0

Discriminant D = b2 – 4ac = 0

[p(1 + 3k)]2 – 4.1.7(3 + 2k) = 0

[ – 2(1 + 3k)]2 – 4.1.7(3 + 2k) = 0

4(1 + 6k + 9k2) – 4.7(3 + 2k) = 0 using (a + b)2 = a2 + 2ab + b2

4(1 + 6k + 9k2 – 21 – 14k) = 0

9k2 – 8k – 20 = 0

9k2 – 18k – 10k – 20 = 0

9k(k – 2) + 10(k – 2) = 0

(9k + 10)(k – 2) = 0

The value of k is for which roots of the quadratic equation are equal.

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