Question

If the roots of the equation \( 3x^2 + 4kx + 3 = 0 \) are non-real, then \( k \) lies in the interval:

• A. \( \left[-2, -\frac{3}{2} \right] \)
• B. \( \left[\frac{3}{2}, 2 \right] \)
• C. \( \left( -\frac{3}{2}, \frac{3}{2} \right) \)
• D. \( (2, 3) \)

Hint:

To determine when a quadratic has non-real roots, always apply the discriminant condition \( b^2 - 4ac<0 \). Then solve the resulting inequality.

Answer 1

The Correct Option is C

We know that the roots are non-real \( \Rightarrow \) discriminant \( D<0 \). \[ D = (4k)^2 - 4ac = 16k^2 - 36<0 \Rightarrow 16k^2<36 \Rightarrow k^2<\frac{9}{4} \] \[ \Rightarrow -\frac{3}{2}<k<\frac{3}{2} \]