# The complete set of values of k, for which the quadratic equation

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.

$\Rightarrow \mathrm{D}=0$

$\Rightarrow \mathrm{b}^{2}-4 \mathrm{ac}=0$

$\Rightarrow \mathrm{k}^{2}-4(1)(\mathrm{k}+2)=0$

$\Rightarrow \mathrm{k}^{2}-4 \mathrm{k}-8=0$

$\Rightarrow \mathrm{k}=\frac{4 \pm \sqrt{16-4(1)(-8)}}{2(1)}$

$\Rightarrow \mathrm{k}=\frac{4 \pm 2 \sqrt{12}}{2}$

$\Rightarrow \mathrm{k}=2 \pm \sqrt{12}$