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. $2-\sqrt{12}$
• C. $-2-\sqrt{12}$
• D. $2\pm \sqrt{12}$

Answer 1

The correct option is D $2\pm \sqrt{12}$

Given: $x^2-kx+k+2=0$ has equal roots

$D=0$

$\Rightarrow b^2-4ac=0$

On comparing with general form of Quadratic Equation

$a{x}^2+bx+c=0$

We get $a=1,b=-k,c=k+2$

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

$\Rightarrow k^2-4k-8=0$

$\Rightarrow k^2-4k+4-12=0$

$\Rightarrow (k-2{)}^2-12=0$

$\Rightarrow (k-2{)}^2=12$

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

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

Hence, the value of $k$ are $2\pm \sqrt{12}$.

Option (B) is correct.