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

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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

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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.

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