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If the equation $x^{2} + 2 (k + 2) x + 9 k = 0$ has equal roots then k = ?

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1 or 4

-1 or -4

1 or -4

-1 or 4

answer is A.

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Detailed Solution

Given quadratic equation is,

x^{2} + 2 (k + 2) x + 9 k = 0

.
We know that a quadratic equation

a x^{2} + bx + c = 0

has two equal roots if

b^{2} - 4 ac = 0

.
After comparing the given equation

x^{2} + 2 (k + 2) x + 9 k = 0

with the standard form of the quadratic equation, we get,

a = 1, b = 2 (k + 2), c = 9 k

.
Then,

⟹ b^{2} - 4 ac = 0

⟹ {[2 (k + 2)]}^{2} - 4 \times 1 \times 9 k = 0

⟹ {(2 k + 4)}^{2} - 36 k = 0

⟹ 4 k^{2} + 16 k + 16 - 36 k = 0

⟹ 4 k^{2} - 36 k + 16 k + 16 = 0

⟹ 4 k^{2} - 20 k + 16 = 0

⟹ 4 k^{2} - 16 k - 4 k + 16 = 0

⟹ 4 k (k - 16) - 4 (k - 4) = 0

⟹ (4 k - 4) (k - 4) = 0

⟹ 4 k - 4 = 0 or k - 4 = 0

⟹ 4 (k - 1) = 0 or k = 4

⟹ k - 1 = 0 or k = 4

∴ k = 1 or k = 4

∴

If the equation

x^{2} + 2 (k + 2) x + 9 k = 0

has equal roots then k = 1 or 4.
Hence, the correct option is 1.

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