Question

\(\frac{1}{x^2-3x+2}+\frac{1}{x^2-5x+6}+\frac{1}{x^2-7x+12}+\frac{1}{x^2-9x+20}=\frac{1}{15}\)

Answer 1

ĐKXĐ:\(x\ne1;2;3;4;5\)

\(\Leftrightarrow\frac{1}{x^2-x-2x+2}+\frac{1}{x^2-2x-3x+6}+\frac{1}{x^2-3x-4x+12}+\frac{1}{x^2-4x-5x+20}=\frac{1}{15}\)

\(\Leftrightarrow\frac{1}{\left(x-1\right)\left(x-2\right)}+\frac{1}{\left(x-2\right)\left(x-3\right)}+\frac{1}{\left(x-3\right)\left(x-4\right)}+\frac{1}{\left(x-4\right)\left(x-5\right)}=\frac{1}{15}\)

\(\Leftrightarrow\frac{1}{x-1}-\frac{1}{x-2}+\frac{1}{x-2}-\frac{1}{x-3}+\frac{1}{x-3}-\frac{1}{x-4}+\frac{1}{x-4}-\frac{1}{x-5}=\frac{1}{15}\)

\(\Leftrightarrow\frac{1}{x-1}-\frac{1}{x-5}=\frac{1}{15}\)

\(\Leftrightarrow\frac{15\left(x-5\right)-15\left(x-1\right)}{15\left(x-1\right)\left(x-5\right)}=\frac{\left(x-1\right)\left(x-5\right)}{15\left(x-1\right)\left(x-5\right)}\)

\(\Rightarrow15x-75-15x+15=x^2-6x+5\)

\(\Leftrightarrow x^2-6x+65=0\)

\(\Leftrightarrow\left(x^2-6x+9\right)+56=0\)

\(\Leftrightarrow\left(x-3\right)^2=-56\) (Vô lý)

Vì bình phương một số không thể bằng âm

Vây \(S=\varnothing\)

Answer 2

tao đéo biết

Answer 3

gợi ý bn nhé:
1/(x²-3x+2) = 1/(x-2)  -   1/(x-1)

tương tự nhé

Answer 4

\(\frac{1}{x^2-3x+2}+\frac{1}{x^2-5x+6}+\frac{1}{x^2-7x+12}+\frac{1}{x^2-9x+20}=\frac{1}{15}\)

\(\Leftrightarrow\frac{1}{\left(x-1\right)\left(x-2\right)}+\frac{1}{\left(x-2\right)\left(x-3\right)}+\frac{1}{\left(x-3\right)\left(x-4\right)}+\frac{1}{\left(x-4\right)\left(x-5\right)}=\frac{1}{15}\)

\(\Leftrightarrow\frac{1}{x-1}-\frac{1}{x-2}+\frac{1}{x-2}-\frac{1}{x-3}+\frac{1}{x-3}-\frac{1}{x-4}+\frac{1}{x-4}-\frac{1}{x-5}=\frac{1}{15}\)

\(\Leftrightarrow\frac{1}{x-1}-\frac{1}{x-5}=\frac{1}{15}\)

\(\Leftrightarrow\frac{15\left(x-5\right)}{15\left(x-1\right)\left(x-5\right)}-\frac{15\left(x-1\right)}{15\left(x-1\right)\left(x-5\right)}=\frac{\left(x-1\right)\left(x-5\right)}{15\left(x-1\right)\left(x-5\right)}\)

\(\Leftrightarrow\frac{15x-75}{15\left(x-1\right)\left(x-5\right)}-\frac{15x-15}{15\left(x-1\right)\left(x-5\right)}=\frac{x^2-6x+5}{15\left(x-1\right)\left(x-5\right)}\)

\(\Leftrightarrow15x-75-\left(15x-15\right)=x^2-6x+5\)

\(\Leftrightarrow15x-75-15x+15=x^2-6x+5\)

\(\Leftrightarrow15x-15x+6x-75+15-5-x^2=0\)

\(\Leftrightarrow6x-65-x^2=0\)

\(\Leftrightarrow0=\left(x^2-6x+9\right)+56\)

\(\Leftrightarrow\left(x-3\right)^2=-56\)(vô lý)

Vì\(\left(x-3\right)^2\ge\forall x\)

\(\Rightarrow x=\varnothing\)