Let α, β be the roots of the equation x²-x+2=0 with Im(α)>Im(β). Then α⁶+α⁴+β⁴-5 α² is equal to ___________.

Solution

Maths sir · 4 steps

Do not find the roots at all: keep replacing α² by α - 2 until every power is linear.

1. Reduce the square. A root satisfies its equation, so α² = α - 2, and the same holds for β. Use this to cut every higher power down to the form pα + q.
2. Climb the powers. α³ = α · α² = α² - 2α = -α - 2. Then α⁴ = α · α³ = -α² - 2α = -3α + 2. And α⁶ = (α³)² = α² + 4α + 4 = 5α + 2.
3. Combine. α⁶ + α⁴ - 5α² = (5α + 2) + (-3α + 2) - 5(α - 2) = -3α + 14. In the same way β⁴ = -3β + 2.

   α⁶+α⁴+β⁴-5α² = -3(α+β) + 16

4. So. The sum of the roots is α + β = 1, so the value is -3 + 16 = 13. The condition on Im(α) only names the roots; it does not change the answer.

The common slip: 19. Taking α + β = -1 instead of +1 (the sum is -b/a) gives 3 + 16 = 19, not 13.

The idea to keep. For high powers of a root of a quadratic, use the equation itself to replace x² again and again. Every power becomes linear, and Vieta's sum finishes the job.

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