Question

[an]

\(\frac{2}{\left(x-1\right)\left(x-3\right)}\)+\(\frac{5}{\left(x-3\right)\left(x-8\right)}\)+\(\frac{12}{\left(x-8\right)\left(x-20\right)}\)-\(\frac{1}{x-20}\)=-\(\frac{3}{4}\)

Answer 1

[Le Thi Khanh Huyen]

\(\frac{2}{\left(x-1\right)\left(x-3\right)}+\frac{5}{\left(x-3\right)\left(x-8\right)}+\frac{12}{\left(x-8\right)\left(x-20\right)}-\frac{1}{x-20}=-\frac{3}{4}\)

\(=\frac{1}{x-1}-\frac{1}{x-3}+\frac{1}{x-3}-\frac{1}{x-8}+\frac{1}{x-8}-\frac{1}{x-20}-\frac{1}{x-20}=-\frac{3}{4}\)

\(=\frac{1}{x-1}-\frac{2}{x-20}=-\frac{3}{4}\)

\(\frac{-x-18}{\left(x-1\right)\left(x-20\right)}=-\frac{3}{4}\)

\(\frac{-x-18}{x^2-21x+20}=\frac{-3}{4}\)

\(\frac{x+18}{x^2-21x+20}=\frac{3}{4}\)

\(4\left(x+18\right)=3\left(x^2-21x+20\right)\)

\(4x+72=3x^2-63x+60\)

\(3x^2-63x-4x=72-60\)

\(3x^2-67x=12\)

\(x\left(2x-67\right)=12\)

\(\Rightarrow x;2x-67\inƯ\left(12\right)=\left\{-12;-6;-4;-3;-2;-1;1;2;3;4;6;12\right\}\)

Mà 2x - 67 lẻ.

Ta có bảng sau:

2x-67	2x	x	(2x - 67 ) . x
-3	64	32	-96 ( loại)
-1	66	33	-33 ( loại )
1	68	34	34 ( loại)
3	70	35	105(loại)

Do đó không có \(x\)thỏa mãn.