Question

For which one of the following values of \( k \), the equation \[ 2x^3 + 3x^2 - 12x - k = 0 \] has three distinct real roots?

• A. 16
• B. 20
• C. 26
• D. 31

Hint:

To determine the number of real roots for a cubic equation, calculate its discriminant. If the discriminant is positive, the equation has three distinct real roots.

Answer 1

The Correct Option is A

Step 1: Analyze the cubic equation.
The equation \( 2x^3 + 3x^2 - 12x - k = 0 \) is a cubic equation. A cubic equation can have at most three real roots, and the number of distinct real roots depends on the discriminant of the cubic equation. The discriminant is a function of the coefficients of the equation, and it can help us determine when the equation has three distinct real roots.
Step 2: Find the discriminant for different values of \( k \).
By solving or using numerical methods, we find that for \( k = 20 \), the cubic equation has three distinct real roots.

Step 3: Conclusion.
The correct value of \( k \) for which the equation has three distinct real roots is \( \boxed{(B)} \).