Question

The complete set of values of k, for which the quadratic equation $x^2-kx+k+2=0$ has equal roots, consists of
(a) $2+\sqrt{12}$
(b) $2\pm \sqrt{12}$
(c) $2-\sqrt{12}$
(d) $-2-\sqrt{12}$

Answer 1

(b) $2\pm \sqrt{12}$


$$\begin{gathered} \mathrm{Since}\mathrm{the}\mathrm{equation}\mathrm{has}\mathrm{real}\mathrm{roots}. \\ \Rightarrow \mathrm{D}=0 \\ \Rightarrow {\mathrm{b}}^2-4\mathrm{ac}=0 \\ \Rightarrow {\mathrm{k}}^2-4\left(1\right)\left(\mathrm{k}+2\right)=0 \\ \Rightarrow {\mathrm{k}}^2-4\mathrm{k}-8=0 \\ \Rightarrow \mathrm{k}=\frac{4\pm \sqrt{16-4\left(1\right)\left(-8\right)}}{2\left(1\right)} \\ \Rightarrow \mathrm{k}=\frac{4\pm 2\sqrt{12}}{2} \\ \Rightarrow \mathrm{k}=2\pm \sqrt{12} \end{gathered}$$