3) Упростите выражение a) 3x/x-3 + x+5/6-2x + 54/5x+x²; б) 48x/16-x²:(x+4/x-4 - x-4/x+4)

a)

$$\frac{3x}{x-3} + \frac{x+5}{6-2x} + \frac{54}{5x+x^2} = \frac{3x}{x-3} - \frac{x+5}{2(x-3)} + \frac{54}{x(x+5)} = \frac{3x \cdot 2x(x+5) - (x+5)x(x+5) + 54 \cdot 2(x-3)}{2x(x-3)(x+5)} = \frac{6x^3+30x^2 - (x^3+10x^2+25x) + 108x - 324}{2x(x-3)(x+5)} = \frac{5x^3 + 20x^2 + 83x - 324}{2x(x-3)(x+5)}$$ Ответ: $$\frac{5x^3 + 20x^2 + 83x - 324}{2x(x-3)(x+5)}$$

б)

$$\frac{48x}{16-x^2} : (\frac{x+4}{x-4} - \frac{x-4}{x+4}) = \frac{48x}{(4-x)(4+x)} : (\frac{(x+4)(x+4) - (x-4)(x-4)}{(x-4)(x+4)}) = \frac{48x}{(4-x)(4+x)} : (\frac{x^2+8x+16 - (x^2-8x+16)}{(x-4)(x+4)}) = \frac{48x}{(4-x)(4+x)} : (\frac{16x}{(x-4)(x+4)}) = \frac{48x}{-(x-4)(x+4)} \cdot \frac{(x-4)(x+4)}{16x} = \frac{48x(x-4)(x+4)}{-16x(x-4)(x+4)} = -3$$ Ответ: -3