Question

Решить дробно рациональное уравнение
x-6/x+2=x+8/x-4
вычислить ОДЗ или выполнить проверку корней

Answer 1

$\frac{x - 6}{x + 2} = \frac{x + 8}{x - 4}$

$\frac{x - 6}{x + 2} = \frac{x +8 }{x - 4} ,x≠ - 2,x = 4$

$(x - 6) \times (x - 4) = (x + 6) \times (x + 2)$

$x {}^{2} - 4x - 6x + 24 = x {}^{2} + 2x + 8x + 16$

$- 4x - 6x + 24 = 2x + 8x + 16$

$- 10x + 24 = 10x + 16$

$- 10x - 10x = 16 - 24$

$- 20x = - 8$

$x = \frac{2}{5} ,x≠ - 2,x≠4$

$x = \frac{2}{5}$

$Ответ:x = \frac{2}{5}$

Альтернативная форма:

$x = 0.4$

Answer 2

$\displaystile\bf\\\frac{x-6}{x+2} =\frac{x+8}{x-4} \\\\\\\frac{x-6}{x+2} -\frac{x+8}{x-4} =0\\\\\\\frac{(x-6)\cdot(x-4)-(x+8)\cdot(x+2)}{(x+2)(x-4)} =0\\\\\\\frac{x^{2} -4x-6x+24-x^{2} -2x-8x-16}{(x+2)(x-4)} =0\\\\\\\frac{-20x+8}{(x+2)(x-4)}=0\\\\\\\left \{ {{-20x+8=0} \atop {x+2\neq 0 \ ; \ x-4\neq 0}} \right. \\\\\\\left \{ {{x=0,4} \atop {x\neq -2 \ ; \ x\neq 4}} \right. \\\\\\Otvet:0,4$