Question

find the value of k for which the quadratic equation 3x²-2x-k=0 has two real roots​

Answer 1

Solution:

Given equation,

→ 3x² - 2x - k = 0

Comparing the given equation with ax² + bx + c = 0, we get,

→ a = 3

→ b = -2

→ c = -k

Now calculate the discriminant of the given equation,

→ D = b² - 4ac

‎→ D = (-2)² - 4 × 3 × (-k)

→ D = 4 + 12k

As the roots are real,

→ D ≥ 0

→ 4 + 12k ≥ 0

→‎ 4(1 + 3k) ≥ 0

Dividing both sides by 4, we get,

→ 3k + 1 ≥ 0

→ 3k + 1 - 1 ≥ 0 - 1

→ 3k ≥ - 1

‎→ 3k/3 ≥ -1/3

→ k ≥ -1/3, k ∈ R.

★ Hence, for any values of k ≥ -1/3, the roots of the given equation are real.

Know More:

Discriminant: The discriminant of any equation tells us about the nature of roots.

The general form of a quadratic equation is -

→ ax² + bx + c = 0

Discriminant is calculated by using the formula -

→ D = b² - 4ac

1. When D > 0: Roots are real and distinct.

2. When D < 0: Roots are imaginary.

3. When D = 0: Roots are real and equal.