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calculati a la puterea 2 + 1/a la 2,stiind ca a la 2 -4a +1 =o , adiferit de 0

Răspuns :

ModFriendly

[tex] a^2-4a+1=0 \ |:a, \ a \neq 0\\ \\ a-4+\frac{1}{a}=0 \Rightarrow a+\frac{1}{a}=4 \ |^2\\ \\\Rightarrow (a+\frac{1}{a})^2=4^2 \\ \\ \Rightarrow a^2+2\cdot a \cdot \frac{1}{a}+\frac{1}{a^2}=16\\ \\ \Rightarrow a^2+2\cdot a +\frac{1}{a^2}=16\\ \\ \Rightarrow a^2+\frac{1}{a^2}=16-2\\ \\ \Rightarrow \boxed{a^2+\frac{1}{a^2}=14}[/tex]

Nicumavro

Răspuns:

Explicație pas cu pas:

solutia 1

ecuatia a²-4a+1=0 se mai scrie

(a-2)²-3=0

a-2=+/-√3

a1= 2-√3   a²=4-4√3+3=7-4√3

a2= 2+√3   a²=4+4√3+3=7+4√3

deci ptr a1:

a²+1/a²=(7-4√3)+1/(7-4√3)=7-4√3+(7+4√3)/(49-48)=14 (am rationalizat numitorul)

ptr a2: (7+4√3)+1/(7+4√3)=7+4√3+(7-4√3)/(49-48)=14

SOL. II

a²-4a+1=0

a²=4a-1

a²+1/a²= (4a-1)+1/(4a-1)=(16a²-8a+1)/(4a-1)=(16a²-8a+1)/a²=16-2(4a-1)/a²=16-2a²/a²=16-2=14

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