Question

.......................​

Answer 1

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