Question

What is the nature of the roots of the quadratic equation \( \frac{4}{3}x^2 - 2x + \frac{3}{4} = 0 \)?

• A. Real and unequal
• B. Real and equal
• C. Not real
• D. None of these

Hint:

The nature of roots depends on \( b^2 - 4ac \): \[ \begin{cases} > 0, & \text{Real and unequal} \\ = 0, & \text{Real and equal} \\< 0, & \text{Not real} \end{cases} \]

Answer 1

The Correct Option is B

The nature of the roots depends on the discriminant:
\[ b^2 - 4ac. \] For \( a = \frac{4}{3}, b = -2, c = \frac{3}{4} \): \[ (-2)^2 - 4 \times \frac{4}{3} \times \frac{3}{4}. \] \[ 4 - \left(\frac{16}{3} \times \frac{3}{4} \right). \] \[ 4 - \frac{48}{12} = 4 - 4 = 0. \] Since the discriminant is zero, the roots are real and equal.