Question:
The complete set of values of $k$, for which the quadratic equation $x^{2}-k x+k+2=0$ has equal roots, consists of
(a) $2+\sqrt{12}$
(b) $2 \pm \sqrt{12}$
(c) $2-\sqrt{12}$
(d) $-2-\sqrt{12}$
Solution:
(b) $2 \pm \sqrt{12}$
Since the equation has real roots.
$\Rightarrow \mathrm{D}=0$
$\Rightarrow \mathrm{b}^{2}-4 \mathrm{ac}=0$
$\Rightarrow \mathrm{k}^{2}-4(1)(\mathrm{k}+2)=0$
$\Rightarrow \mathrm{k}^{2}-4 \mathrm{k}-8=0$
$\Rightarrow \mathrm{k}=\frac{4 \pm \sqrt{16-4(1)(-8)}}{2(1)}$
$\Rightarrow \mathrm{k}=\frac{4 \pm 2 \sqrt{12}}{2}$
$\Rightarrow \mathrm{k}=2 \pm \sqrt{12}$
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