Correct Answer - Option 2 : β = 4α

^{2} - 3α

__Concept:__

General Quadratic Equation

ax2 + bx + c = 0

- Product of roots (αβ) = c/a
- Sum of roots (α + β) = -b/a
- Root find by using this formula, \(\rm x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
- sin 3θ = 3sinθ - 4sin
^{3}θ

**Calculation:**

Given: 4x2 + 2x - 1 = 0

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

⇒ \(x = {-2 \pm \sqrt{2^2-4(4)(-1)} \over 2(4)}\)

⇒ \(α = {-1 + \sqrt{5} \over 4}\) and \(β = {-1 - \sqrt{5} \over 4}\)

As we know that \(sin18° = {-1 + \sqrt{5} \over 4}\) and \(sin54° = {1 + \sqrt{5} \over 4}\)

So, We can say that α = sin18° and β = - sin54° -----(i)

Now, From the given formula

sin 3θ = 3sinθ - 4sin3θ

On putting θ = 18° in the above trigonometric formula, we get

⇒ sin 54° = 3sin18° - 4sin318° ----(ii)

From (i) and (ii), we get

⇒ -β = 3α - 4α^{3}

⇒ β = 4α2 - 3α

**∴ The correct relation is β = 4α2 - 3α.**