Question

The complete set of values of k, for which the quadratic equation  \[x^2 - kx + k + 2 = 0\] has equal roots, consists of

Options

• \[2 + \sqrt{12}\]

• \[2 \pm \sqrt{12}\]

• \[2 - \sqrt{12}\]

• \[- 2 - \sqrt{12}\]

Answer 1

\[2 \pm \sqrt{12}\]

\[\text { Since the equation has real roots } . \]

\[ \Rightarrow D = 0\]

\[ \Rightarrow b^2 - 4ac = 0\]

\[ \Rightarrow k^2 - 4\left( 1 \right)\left( k + 2 \right) = 0\]

\[ \Rightarrow k^2 - 4k - 8 = 0\]

\[ \Rightarrow k = \frac{4 \pm \sqrt{16 - 4\left( 1 \right)\left( - 8 \right)}}{2\left( 1 \right)}\]

\[ \Rightarrow k = \frac{4 \pm 2\sqrt{12}}{2}\]

\[ \Rightarrow k = 2 \pm \sqrt{12}\]