Question

If roots of the quadratic equation \(x^2 - k\sqrt{3}x + 2 = 0\) are real and equal, then value of \(k\) is

• A. \(-2\)
• B. \(\sqrt{\frac{8}{3}}\)
• C. \(1\)
• D. \(2\)

Hint:

When a quadratic has "equal roots", it's a perfect square trinomial. The middle coefficient squared always equals \(4 \times \text{first term} \times \text{last term}\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For a quadratic equation \(ax^2 + bx + c = 0\) to have real and equal roots, its discriminant must be zero.
Step 2: Key Formula or Approach:
Discriminant \(D = b^2 - 4ac = 0\).
Here, \(a = 1, b = -k\sqrt{3}, c = 2\).
Step 3: Detailed Explanation:
Substitute values into the discriminant formula:
\[ (-k\sqrt{3})^2 - 4(1)(2) = 0 \]
\[ 3k^2 - 8 = 0 \]
\[ 3k^2 = 8 \]
\[ k^2 = \frac{8}{3} \]
\[ k = \pm \sqrt{\frac{8}{3}} \]
Comparing with options, (B) provides the positive value.
Step 4: Final Answer:
The value of \(k\) is \(\sqrt{\frac{8}{3}}\).