Let α and β be two roots of the equation x² + 2x + 2 = 0. Then α₁₅ + β₁₅ is equal to

• A: 512
• B: -512
• C: -256
• D: 256

To find the value of α₁₅ + β₁₅ for the roots of the equation x² + 2x + 2 = 0, we first need to determine the roots α and β. We can use the quadratic formula:

Finding the Roots

The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

For our equation, a = 1, b = 2, and c = 2. Plugging in these values:

x = (-2 ± √(2² - 4 × 1 × 2)) / (2 × 1)

This simplifies to:

x = (-2 ± √(4 - 8)) / 2

x = (-2 ± √(-4)) / 2

Thus, we have:

x = -1 ± i

So, the roots are:

• α = -1 + i
• β = -1 - i

Calculating α₁₅ + β₁₅

To find α₁₅ + β₁₅, we can use the property of roots of unity. The roots can be expressed in exponential form:

α = √2 (cos(3π/4) + i sin(3π/4))

β = √2 (cos(5π/4) + i sin(5π/4))

Using De Moivre's Theorem

According to De Moivre's theorem:

α_n + β_n = 2 * (√2)ⁿ * cos(nθ)

For n = 15, we have:

α₁₅ + β₁₅ = 2 * (√2)¹⁵ * cos(15 * 3π/4)

Calculating this gives:

α₁₅ + β₁₅ = 2 * 2^7.5 * cos(15 * 3π/4)

Since cos(15 * 3π/4) = -√2/2, we find:

α₁₅ + β₁₅ = 2 * 2^7.5 * (-√2/2)

This simplifies to:

α₁₅ + β₁₅ = -2⁸ = -256

Final Answer

The value of α₁₅ + β₁₅ is -256. Therefore, the correct option is:

C: -256