Question

If α and β are the roots of the equation 4x² + 2x - 1 = 0, then which one of the following is correct?
1. β = -2α² - 2α
2. β = 4α² - 3α
3. β = α² - 3α
4. β = -2α² + 2α

Answer 1

Correct Answer - Option 2 : β = 4α² - 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 - 4sin³θ 

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 - 4α³

⇒ β = 4α2 - 3α

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