Question

$x - 2/x²+1 + x²+1/x-2 = -5/2$
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Answer 1

Răspuns:

Explicație pas cu pas:

$\dfrac{x-2}{x^{2}+1}+\dfrac{x^{2}+1}{x-2}=-\dfrac{5}{2}\\Fie~t=\dfrac{x-2}{x^{2}+1},~~atunci~\dfrac{x^{2}+1}{x-2}=\dfrac{1}{t} .~Obtinem~ecuatia~~~t+\dfrac{1}{t}=-\dfrac{5}{2}~|*2t$

Obtinem, 2t²+5t+2=0, Δ=5²-4·2·2=25-16=9

t1=(-5-3)/(2·2)=-2, iar t2=(-5+3)/(2·2)=-1/2. Revenim la variabila x

$\dfrac{x-2}{x^{2}+1} =-2,$   ⇒x-2=-2(x²+1) ⇒ x-2=-2x²-2⇒ 2x²+x=0 ⇒ x(2x+1)=0 deci x=0 sau x=-1/2.

$\dfrac{x-2}{x^{2}+1}=\dfrac{-1}{2}$   ⇒2(x-2)=-1(x²+1) ⇒ 2x-4=-x²-1 ⇒ x²+2x-3=0, Δ=4+12=16

x=(-2-4)/2=-3 sau x=(-2+4)/2=1

Raspuns:  S={-3; -1/2; 0; 1}