MCQ
If the equation $x^2+ 6(k + 2)x + 9k = 0$ has equal roots then $k = ?$
- ✓$1$ or $4$
- B$-1$ or $4$
- C$1$ or $-4$
- D$-1$ or $-4$
✓
Answer
Correct option: A.
$1$ or $4$
Since the roots of the equation $x^2+ 6(k + 2)x + 9k = 0$ are equal,
$D = 0$
$ \Rightarrow b^2-4 a c=0 $
$ \Rightarrow[2(k+2)]^2-4 \times 1 \times 9 k=0 $
$ \Rightarrow 4\left(k^2+4 k+4\right)-36 k=0 $
$ \Rightarrow 4 k^2+16 k+16-36 k=0 $
$ \Rightarrow 4 k^2-20 k+16=0 $
$ \Rightarrow k^2-5 k+4=0 $
$ \Rightarrow k^2-4 k-k+4=0 $
$\Rightarrow k(k - 4) -1(k - 4) = 0$
$\Rightarrow (k - 4)(k - 1) = 0$
$\Rightarrow k - 4 = 0$ or $k - 1 = 0$
$\Rightarrow k = 4$ or $k = 1$
$D = 0$
$ \Rightarrow b^2-4 a c=0 $
$ \Rightarrow[2(k+2)]^2-4 \times 1 \times 9 k=0 $
$ \Rightarrow 4\left(k^2+4 k+4\right)-36 k=0 $
$ \Rightarrow 4 k^2+16 k+16-36 k=0 $
$ \Rightarrow 4 k^2-20 k+16=0 $
$ \Rightarrow k^2-5 k+4=0 $
$ \Rightarrow k^2-4 k-k+4=0 $
$\Rightarrow k(k - 4) -1(k - 4) = 0$
$\Rightarrow (k - 4)(k - 1) = 0$
$\Rightarrow k - 4 = 0$ or $k - 1 = 0$
$\Rightarrow k = 4$ or $k = 1$
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