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:

For real and equal roots, the discriminant \( \Delta = 0 \).

Answer 1

The Correct Option is A

We need to determine the nature of the roots of the quadratic equation \( \frac{4}{3}x^2 - 2x + \frac{3}{4} = 0 \). The nature of the roots depends on the discriminant \( \Delta \), given by: \[ \Delta = b^2 - 4ac. \] For the equation \( \frac{4}{3}x^2 - 2x + \frac{3}{4} = 0 \), we have \( a = \frac{4}{3} \), \( b = -2 \), and \( c = \frac{3}{4} \). The discriminant is: \[ \Delta = (-2)^2 - 4 \times \frac{4}{3} \times \frac{3}{4} = 4 - 4 = 0. \] Since the discriminant is zero, the roots are real and equal. Thus, the roots are \( \boxed{\text{Real and equal}} \).