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:

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.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The topic of this question is Quadratic Equations.
A quadratic equation is of the standard form \(ax^2 + bx + c = 0\), where \(a \neq 0\).
The nature of the roots of a quadratic equation depends on its discriminant, denoted by \(D\).
For a quadratic equation to have real and equal roots, the discriminant must be exactly equal to zero.
We are given the quadratic equation \(x^2 - k\sqrt{3}x + 2 = 0\) and we need to determine the value of the constant \(k\).

Step 2: Key Formula or Approach:
The discriminant \(D\) of the quadratic equation \(ax^2 + bx + c = 0\) is given by the formula:
\[ D = b^2 - 4ac \] For real and equal roots, the condition is:
\[ D = 0 \implies b^2 - 4ac = 0 \] We will identify the coefficients from the given equation, substitute them into the discriminant formula, and solve for \(k\).

Step 3: Detailed Explanation:

• Compare the given equation \(x^2 - k\sqrt{3}x + 2 = 0\) with the standard form \(ax^2 + bx + c = 0\):
Here, the coefficients are:
\(a = 1\)
\(b = -k\sqrt{3}\)
\(c = 2\)

• Substitute these coefficients into the discriminant formula:
\[ D = (-k\sqrt{3})^2 - 4(1)(2) \]

• Simplify the expression:
The square of \(-k\sqrt{3}\) is \(3k^2\), and the product of \(4\), \(1\), and \(2\) is \(8\).
\[ D = 3k^2 - 8 \]

• Set the discriminant equal to zero for real and equal roots:
\[ 3k^2 - 8 = 0 \] \[ 3k^2 = 8 \]

• Solve for \(k^2\):
\[ k^2 = \frac{8}{3} \]

• Take the square root on both sides to find \(k\):
\[ k = \pm \sqrt{\frac{8}{3}} \] Since only the positive root is provided in the choices, we select \(\sqrt{\frac{8}{3}}\).


Step 4: Final Answer:
The value of \(k\) for which the quadratic equation has real and equal roots is \(\sqrt{\frac{8}{3}}\).
Therefore, the correct option is (B).