Question

Find the nature of roots of the equation \(3x^2 - 4\sqrt{3}x + 4 = 0\).

Answer 1

Quadratic Equation: Nature of the Roots

The given quadratic equation is:

\[ 3x^2 - 4\sqrt{3}x + 4 = 0 \]

This is in the standard form \( ax^2 + bx + c = 0 \), where:

• \( a = 3 \)
• \( b = -4\sqrt{3} \)
• \( c = 4 \)

Step 1: Calculate the Discriminant

\[ D = b^2 - 4ac = (-4\sqrt{3})^2 - 4(3)(4) \] \[ = (16)(3) - 48 = 48 - 48 = 0 \]

Step 2: Interpret the Discriminant

Since \( D = 0 \), the quadratic equation has: Real and Equal Roots.

✅ Conclusion: The roots of the equation are real and equal.