Let α, β be the roots of the equation x²-√2 x+2=0. Then α¹⁴+β¹⁴ is equal to

Solution

Maths sir · 4 steps

Write both roots in polar form, then De Moivre turns the 14th power into a quick angle calculation.

1. Find the roots. By the quadratic formula, x = (√2 ± √(2 - 8))/2 = (√2 ± i√6)/2. The two roots are complex conjugates.
2. Polar form. The modulus is √(2/4 + 6/4) = √2. Taking √2 out gives α, β = √2(1/2 ± i√3/2) = √2 e^(± iπ/3).

   α¹⁴ + β¹⁴ = (√2)¹⁴(e^(i 14π/3) + e^(-i 14π/3)) = 128 × 2 cos 14π/3

3. Reduce the angle. 14π/3 = 4π + 2π/3, so cos 14π/3 = cos 2π/3 = -1/2. Also (√2)¹⁴ = 2⁷ = 128.
4. So. α¹⁴ + β¹⁴ = 128 × 2 × (-1/2) = -128, which is option (d).

The idea to keep. For conjugate roots re^(± iθ), the sum of nth powers is 2rⁿ cos nθ. Find r and θ first, then remove full turns of 2π from nθ.

Answer from the NTA final answer key. Solution by PrepWiser.