Question

Find the value of \( k \) for which the quadratic equation \( 2x^2 + kx + 3 = 0 \) has two equal real roots.

Hint:

For equal roots in a quadratic equation, always set the discriminant \( b^2 - 4ac = 0 \). This is the fastest method in exams.

Answer 1

Concept: A quadratic equation \( ax^2 + bx + c = 0 \) has:

• Two equal real roots when the discriminant \( D = 0 \)
• Where \( D = b^2 - 4ac \)

Thus, we use the discriminant condition to determine \( k \).
Step 1: Identify coefficients.
Given equation: \[ 2x^2 + kx + 3 = 0 \] So, \[ a = 2, \quad b = k, \quad c = 3. \]
Step 2: Use the condition for equal roots.
For equal real roots: \[ D = b^2 - 4ac = 0. \]
Step 3: Substitute values.
\[ k^2 - 4(2)(3) = 0 \] \[ k^2 - 24 = 0. \]
Step 4: Solve for \( k \).
\[ k^2 = 24 \] \[ k = \pm \sqrt{24} = \pm 2\sqrt{6}. \] Conclusion:
The quadratic equation has equal real roots when: \[ k = \pm 2\sqrt{6}. \]