Find the greatest value of k ∈ N for which the equation x² - 4x + k = 0 has distinct real roots.

1. -4

2. 3

3. 1

4. 4

Given,

x² - 4x + k = 0

Comparing x² - 4x + k = 0 with ax² + bx + c = 0 we get,

a = 1, b = -4 and c = k.

Since roots are distinct and real,

∴ D > 0

⇒ b² - 4ac > 0

⇒ (-4)² - 4(1)(k) > 0

⇒ 16 - 4k > 0

⇒ 16 > 4k

⇒ $\dfrac{16}{4}$ > k

⇒ k < 4

Since k ∈ N,

Possible natural numbers less than 4 are: 1, 2, 3

The greatest value is: 3

Hence, Option 2 is the correct option.