If roots of the quadratic equation \(x^2 - k\sqrt{3}x + 2 = 0\) are real and equal, then value of k is
Show Hint
Always isolate the square term before taking the square root.
If \(k^2 = \frac{8}{3}\), then \(k = \pm \sqrt{\frac{8}{3}}\).
Since only the positive root is listed in the options, select that value directly.
Step 1: Try completing the square instead of using the discriminant formula.
Write $x^2 - k\sqrt{3}x + 2$ as $\left(x - \frac{k\sqrt{3}}{2}\right)^2 + \left(2 - \frac{3k^2}{4}\right)$. Step 2: Set the leftover constant term to zero.
For the equation to reduce to a perfect square (equal roots), the term outside the square bracket must vanish:
\[ 2 - \frac{3k^2}{4} = 0 \] Step 3: Solve for k.
\[ \frac{3k^2}{4} = 2 \implies 3k^2 = 8 \implies k^2 = \frac{8}{3} \implies k = \sqrt{\frac{8}{3}} \]
\[ \boxed{k = \sqrt{\frac{8}{3}}} \]