If the equation x 2 + 2 ( k + 2 ) x + 9 = 0 has equal roots, then find the values of k. Options: A . &nbsp 1, 4 B . &nbsp –1, 5 C . &nbsp 1, –5 D . &nbsp –1, –4

If the equation x2+2(k+2)x+9=0 has equal roots, then find the values of k.

Question

If the equation $x^{2} + 2 (k + 2) x + 9 = 0$ has equal roots, then find the values of k.

Options:

A . 1, 4

B . –1, 5

C . 1, –5

D . –1, –4

Answer: Option C
:
C

Step 1:- $x^{2} + 2 (k + 2) x + 9 = 0$ , $\Rightarrow a = 1, b = 2 (k + 2), c = 9$

$D = 4 (k + 2)^{2} - 4 (9) = 4 (k^{2} + 4 k - 5)$ [ $D = b^{2} - 4 a c$ ]

Step 2:- The roots of quadratic equation are real and equal only when $D = 0$

$4 (k^{2} + 4 k - 5) = 0, \Rightarrow k^{2} + 4 k - 5 = 0$

Step 3:- $k^{2} + 4 k - 5 = 0$

$\Rightarrow k^{2} + 5 k - k - 5 = 0$

$\Rightarrow k (k + 5) - 1 (k + 5) = 0$

$\Rightarrow (k - 1) (k + 5) = 0$

$\Rightarrow k = 1 o r k = - 5$

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