y² = x + 1 (1)
y = 2x² / 9 (2)
由(1)代入(2)式:
(2x² / 9)² = x + 1
得 4x^4 - 81x - 91 = 0
利用余式定理,易得
方程可分解成
(x - 3)(4x^3 + 12x² + 36x + 27) = 0
x = 3 或 4x^3 + 12x² + 36x + 27 = 0
在方程4x^3 + 12x² + 36x + 27 = 0中,(以下采用卡丹公式来做)
令x^3系数为1,x^3 + 3x² + 9x + 27/4 = 0
设 x = y - 1 (消除x²项系数)
则方程化为 y^3 + 6y - 1/4 = 0
∴ p = 6,q = -1/4
∵ D = q²/4 + p^3 / 27
= (-1/4)² / 4 + 6^3 / 27
= 513 / 64 > 0
∴此方程只有一个实数根,(另两根为共轭复数)
由 y = (-q/2 + √D)^(1/3) + (-q/2 - √D)^(1/3) (代q = -1/4,D = 513 / 64)
= [(1 + 3√57)^(1/3) + (1 - 3√57)^(1/3)] / 2
∴ x = y - 1 = [(1 + 3√57)^(1/3) + (1 - 3√57)^(1/3)] / 2 - 1
所以两方程的解为
x = 3,
y = 2
或
x = [(1 + 3√57)^(1/3) + (1 - 3√57)^(1/3)] / 2 - 1,
y = √ {[(1 + 3√57)^(1/3) + (1 - 3√57)^(1/3)] / 2 }