Question

Giải phương trình: \(\frac{1}{\left(x^2+x+1\right)^2}+\frac{1}{\left(x^2+x+2\right)^2}=\frac{5}{4}\)

Answer 1

Đặt \(x^2+x+1=a\)

\(pt\Leftrightarrow\frac{1}{a^2}+\frac{1}{\left(a+1\right)^2}=\frac{5}{4}.\)

\(\Leftrightarrow\left(\frac{1}{a}-\frac{1}{a+1}\right)^2+\frac{2}{a\left(a+1\right)}-\frac{5}{4}=0\)

\(\Leftrightarrow\left(\frac{1}{a\left(a+1\right)}\right)^2+\frac{2}{a\left(a+1\right)}-\frac{5}{4}=0\)

đặt \(\frac{1}{a\left(a+1\right)}=b\)

\(\Leftrightarrow b^2+2b-\frac{5}{4}=0\Leftrightarrow4b^2+8b-5=0\)

\(\left(2b-1\right)\left(2b+5\right)=0.\)

đến đây tự full đi.