Question

Giải phương trình \(\dfrac{x^2+x}{x^2+3}-\dfrac{3x^2-x+15}{x^2+4}+\dfrac{x^2+x+2}{x^2+5}+x^3-3x^2+1=0\)

Answer 1

Từ đề suy ra \(\left(\dfrac{x^2+x}{x^2+3}-1\right)+\left(3-\dfrac{3x^2-x+15}{x^2+4}\right)+\left(\dfrac{x^2+x+2}{x^2+5}-1\right)+x^3-3x^2=0\)

\(\Leftrightarrow\dfrac{x-3}{x^2+3}+\dfrac{x-3}{x^2+4}+\dfrac{x-3}{x^2+5}+x^2\left(x-3\right)=0\)

\(\Leftrightarrow\left(x-3\right)\left(\dfrac{1}{x^2+3}+\dfrac{1}{x^2+4}+\dfrac{1}{x^2+5}+x^2\right)=0\)

=> x - 3 = 0 do \(\dfrac{1}{x^2+3}+\dfrac{1}{x^2+4}+\dfrac{1}{x^2+5}+x^2>0\forall x\)

=> x = 3