Question

已知:X=(1-√2022)/2,求(4X³-2025X-2022)³的ŀ

Answer 1

两边同乘2：2x=1 - √2022
两边平方：(2x)²=(1 - √2022)²
4x²=1 - 2√2022 + 2022
∴4x²=2023 - 2√2022
则4x³ - 2025x - 2022=4x²•x - 2025x - 2022
=(2023 - 2√2022)x - 2025x - 2022
=2023x - (2√2022)x - 2025x - 2022
=-2x - (2√2022)x - 2022
=-2x(1 + √2022) - 2022
=-(1 - √2022)(1 + √2022) - 2022
=-[1² - (√2022)²] - 2022
=-(1 - 2022) - 2022
=-1 + 2022 - 2022
=-1
∴原式=(-1)³=-1

Answer 2

[x-(1-√2022)/2]*[x-(1+√2022)/2]
=x^2-x-2021/4=0,
所以x^2=x+2021/4,
4x^3-2025x-2022
=x(4x^2-2025)-2022
=x(4x-4)-2022
=4x^2-4x-2022
=2021-2022
=-1,
于是(4x^3-2025x-2022)^3=(-1)^3=-1.