Question

If the equation 3x²+2x+k=0 has real roots then k is

• A. \(k < \frac{1}{3}\)
• B. \(k > \frac{1}{3}\)
• C. \(k \leq \frac{1}{3}\)
• D. \(k \geq \frac{1}{3}\)

Answer 1

The Correct Option is C

We are given the quadratic equation: \( 3x^2 + 2x + k = 0 \)

Step 1: For the quadratic equation \( ax^2 + bx + c = 0 \) to have real roots, its discriminant must be greater than or equal to zero.
The discriminant \( D \) is given by:
\( D = b^2 - 4ac \)

Step 2: Compare with the general form:
\( a = 3, \quad b = 2, \quad c = k \)
So, the discriminant becomes:
\( D = 2^2 - 4(3)(k) = 4 - 12k \)

Step 3: For real roots, \( D \geq 0 \):
\( 4 - 12k \geq 0 \)
⇒ \( 12k \leq 4 \)
⇒ \( k \leq \dfrac{1}{3} \)

Therefore, the correct condition is (C) : \( k \leq \dfrac{1}{3} \)