Correct Answer - Option 4 : None of these
Concept:
Consider a quadratic equation ax2 + bx + c = 0
Sum of roots = \(\rm \frac{-b}{a}\)
Products of roots = \(\rm \frac{c}{a}\)
(α + β)2 = α2 + β2 + 2αβ
(α - β)2 = α2 + β2 - 2αβ
α2 - β2 = (α - β) (α + β)
Calculation:
Given: x2 - 2x + 4 = 0
Here a = 1, b = - 2, c = 4
Let roots of the equation are α and β.
Sum of roots = \(\rm \frac{-(-2)}{1}\)
α + β = 2
Products of roots = \(\rm \frac{4}{1}\)
α β = 4
α2 + β2 = (α + β)2 - 2αβ = (2)2 - 2(4) = 4 - 8 = - 4
(α - β)2 = α2 + β2 - 2αβ = - 4 - 2(4) = - 12
α - β = \(\rm 2i\sqrt{3}\)
α2 - β2 = \(\rm 4i\sqrt{3}\)
The value of α2 - β2 is \(\rm 4i\sqrt{3}\)