Question

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

• A. $\pm \sqrt{24}$
• B. 0
• C. 4
• D. -5

Hint:

For equal roots, $b^2 = 4ac$.
You can quickly calculate $4ac$ in your head: $4 \times \sqrt{3} \times 2\sqrt{3} = 4 \times 6 = 24$.
So, $k^2 = 24 \implies k = \pm\sqrt{24}$.
Be careful not to forget the $\pm$ sign, as a quadratic equation with a squared term generally yields two possible values for its parameters!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question is from the chapter "Quadratic Equations", specifically concerning the nature of roots.
A quadratic equation $ax^2 + bx + c = 0$ can have real and distinct roots, real and equal roots, or imaginary roots depending on its discriminant.
We are given a quadratic equation containing an unknown parameter $k$, with the condition that its roots are real and equal.
We need to find the value(s) of $k$ that satisfy this condition.

Step 2: Key Formula or Approach:
The nature of the roots of a quadratic equation is determined by the discriminant $D$:
\[ D = b^2 - 4ac \] The algebraic condition for a quadratic equation to have

real and equal roots is that the discriminant must be exactly equal to zero:
\[ D = b^2 - 4ac = 0 \] We will identify the coefficients $a$, $b$, and $c$ from our given equation, substitute them into the discriminant formula, and solve for $k$.

Step 3: Detailed Explanation:

• Write down the given quadratic equation:
\[ \sqrt{3}x^2 - kx + 2\sqrt{3} = 0 \]

• Compare this with the standard quadratic form $ax^2 + bx + c = 0$ to identify the coefficients:

• $a = \sqrt{3}$

• $b = -k$

• $c = 2\sqrt{3}$

• Set the condition for real and equal roots:
\[ D = b^2 - 4ac = 0 \]

• Substitute the values of $a$, $b$, and $c$ into the formula:
\[ (-k)^2 - 4(\sqrt{3})(2\sqrt{3}) = 0 \]

• Simplify each term in the equation:
\[ (-k)^2 = k^2 \] Now, calculate the multiplication term:
\[ 4 \cdot \sqrt{3} \cdot 2\sqrt{3} = 4 \cdot 2 \cdot (\sqrt{3} \cdot \sqrt{3}) = 8 \cdot 3 = 24 \]

• Substitute these back into the equation:
\[ k^2 - 24 = 0 \] \[ k^2 = 24 \]

• Take the square root on both sides to solve for $k$:
\[ k = \pm\sqrt{24} \] Note that this can be simplified as $\pm 2\sqrt{6}$, but since the options contain $\pm\sqrt{24}$, we leave it in this unsimplified radical form.


Step 4: Final Answer:
The values of $k$ for which the equation has real and equal roots are $\pm\sqrt{24}$, which corresponds to Option (A).