Question

Resolva as equações fracionarias.

a) 2/x-3 - 1/4 = 5/x-3 - 1/3

b) x+4/x+9 = x/ x+3

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Answer 1

a)

$\frac{2}{x-3} - \frac{1}{4} = \frac{5}{x-3} - \frac{1}{3} \\ \\ \frac{2\cdot4-1\cdot(x-3)}{(x-3)\cdot4} = \frac{5\cdot3-1\cdot(x-3)}{(x-3)\cdot3}$

$\frac{8-x+3}{4x-12} = \frac{15-x+3}{3x-9} \\ \\ (8-x+3)\cdot(3x-9)=(15-x+3)\cdot(4x-12) \\ \\ (11-x)\cdot(3x-9)=(18-x)\cdot(4x-12) \\ \\ 33x-99-3x^{2}+9x=72x-216-4x^{2}+12x$

$4x^{2}-3x^{2}+33x-12x-72x+9x-99+216=0 \\ \\ x^{2}-42x+117=0$

$\Delta=(-42)^{2}-4(1)(117)=1764-468=1296$

$x= \frac{-(-42)\pm \sqrt{1296} }{2} \\ \\ x= \frac{42\pm36}{2}$

$x'= \frac{42+36}{2} \\ x'= \frac{78}{2} \\ x'=39$

$x''= \frac{42-36}{2} \\ x''= \frac{6}{2} \\ x''= 3$

b)

$\frac{x+4}{x+9} = \frac{x}{x+3} \\ \\ (x+4)\cdot(x+3)=x\cdot(x+9) \\ \\ x^{2}+3x+4x+12=x^{2}+9x \\ \\ x^{2}-x^{2}+3x+4x-9x=-12 \\ \\ -2x=-12 \\ \\ x=6$