Correct Answer:
D. m = 1/3
The correct answer is m = 1/3. For a quadratic equation in the standard form ax² + bx + c = 0 to have equal roots, its discriminant must be equal to zero. The discriminant is given by the formula b² - 4ac. In the given equation, 3x² + 2x + m = 0, we can identify a = 3, b = 2, and c = m. Setting the discriminant to zero allows us to solve for m.
- Substitute the values into the discriminant formula: (2)² - 4(3)(m) = 0.
- Simplify the equation: 4 - 12m = 0.
- Isolate the term with m: 12m = 4.
- Solve for m: m = 4 / 12 = 1/3.
Thus, when m = 1/3, the quadratic equation will have exactly one real root (or two equal real roots).
- Option A, m = 3, is incorrect. If m = 3, the discriminant would be 4 - 12(3) = 4 - 36 = -32. A negative discriminant indicates two distinct complex (non-real) roots, not equal roots.
- Option B, m = -3, is incorrect. If m = -3, the discriminant would be 4 - 12(-3) = 4 + 36 = 40. A positive discriminant indicates two distinct real roots, not equal roots.
- Option C, m = -1/3, is incorrect. If m = -1/3, the discriminant would be 4 - 12(-1/3) = 4 + 4 = 8. A positive discriminant indicates two distinct real roots, not equal roots.