Giải phương trình (x-3/x-2) - (x-2/x-4) = 16/5 Trình bày chi tiết giúp mình!

Đáp án:

$x=\dfrac{81\pm \sqrt{929}}{32}.$

Giải thích các bước giải:

$\dfrac{x-3}{x-2}-\dfrac{x-2}{x-4}=\dfrac{16}{5} (\text{ĐKXĐ: } x \ne 2; x \ne 4)\\ \Leftrightarrow \dfrac{x-3}{x-2}-\dfrac{x-2}{x-4}-\dfrac{16}{5}=0\\ \Leftrightarrow \dfrac{(x-3).5(x-4)}{5(x-4)(x-2)}-\dfrac{(x-2).5(x-2)}{5(x-4)(x-2)}-\dfrac{16(x-4)(x-2)}{5(x-4)(x-2)}=0\\ \Leftrightarrow \dfrac{5(x-3)(x-4)-5(x-2)(x-2)-16(x-4)(x-2)}{5(x-4)(x-2)}=0\\ \Leftrightarrow \dfrac{-16 x^2 + 81 x - 88}{5(x-4)(x-2)}=0\\ \Rightarrow -16 x^2 + 81 x - 88=0\\ \Leftrightarrow 16 x^2 - 81 x+88=0\\ \Leftrightarrow (4x)^2 -2.4x.\dfrac{81}{8} x+\dfrac{6561}{64}-\dfrac{929}{64}=0\\ \Leftrightarrow \left(4x-\dfrac{81}{8}\right)^2-\dfrac{929}{64}=0\\ \Leftrightarrow \left(4x-\dfrac{81}{8}\right)^2-\left(\dfrac{\sqrt{929}}{8}\right)^2=0\\ \Leftrightarrow \left(4x-\dfrac{81}{8}-\dfrac{\sqrt{929}}{8}\right)\left(4x-\dfrac{81}{8}+\dfrac{\sqrt{929}}{8}\right)=0\\ \Leftrightarrow \left(x-\dfrac{81}{32}-\dfrac{\sqrt{929}}{32}\right)\left(x-\dfrac{81}{32}+\dfrac{\sqrt{929}}{32}\right)=0\\ \Leftrightarrow \left(x-\dfrac{81+\sqrt{929}}{32}\right)\left(x-\dfrac{81-\sqrt{929}}{32}\right)=0\\ \Leftrightarrow x=\dfrac{81\pm \sqrt{929}}{32}.$