Răspuns:
a = 1
Explicație pas cu pas:
x = ∛(7+5√2) = a+√2 I³ =>
x³ = 7+5√2 = (a+√2)³ = (a+√2)(a²+2a√2+2) =
= a³+2a²√2+2a+a²√2+4a+2√2 = a³+3a²√2+6a+2√2 =>
a³+3a²√2+6a+2√2 = 7+5√2 =>
a³+3a²√2+6a+2√2-5√2-7 = 0 =>
a³+3a²√2+6a-3√2-7 = 0
a = 1 => 1³+3·1²·√2+6·1-3√2-7 = 1+3√2+6-3√2-7 = 0 , corect
a³+3a²√2+6a-3√2-7 : (a-1) = a²+a(3√2+1)+(3√2+7)
-a³+a²
--------
= a²(3√2+1) +6a-3√2-7
-a²(3√2+1) +a(3√2+1)
--------------------------
= a(3√2+1+6) - 3√2-7
-a(3√2+7) + 3√2+7
----------------------------
= =
a²+a(3√2+1)+(3√2+7) = 0 =>
Δ = (3√2+1)²-4·(3√2+7) = 18+6√2+1-12√2-28 < 0 => a₂,₃ ∈ C =>
a = 1
