✔ 最佳答案
Equation : x² + 4x + 1 = 0.
Also, it's roots are α and β.
For any equation ax² + bx + c = 0,
sum of it's roots = -b/a
product of it's roots = c/a
So, for the given equation: α + β = -(4)/(1) = -4 and αβ = (1)/(1) = 1.
Using (a+b)² = a² + b² + 2ab
--> a² + b² = (a+b)² - 2ab.
Using the above identity: α² + β² = (α + β)² - 2αβ
--> α² + β² = (-4)² - 2(1)
--> α² + β² = 16 - 2 = 14.
Hence α² + β² = 14.
For α^3 + β^3, using (a + b)^3 = a^3 + b^3 + 3a²b + 3ab²
--> (α + β)^3 - (3a²b + 3ab²) = a^3 + b^3
--> a^3 + b^3 = (a + b)^3 - 3ab(a + b)
Using this, α^3 + β^3 = (α + β)^3 - 3αβ(α + β)
--> α^3 + β^3 = (-4)^3 - 3(1)(-4)
--> α^3 + β^3 = -64 + 12 = -52.
Hence, α^3 + β^3 = -52.