Riješi (x-2/x^2+2x-x-1/x^2+4x+4/frac{4-x{x}})

$$( \frac{ x-2 }{ { x }^{ 2 } +2x } - \frac{ \frac{ x-1 }{ { x }^{ 2 } +4x+4 } }{ \frac{ 4-x }{ x } } )$$

Procijeni

$$\frac{-2x^{3}+5x^{2}+4x-16}{x\left(4-x\right)\left(x+2\right)^{2}}$$

Kratki koraci rješenja

Podijelite \frac{x-1}{x^{2}+4x+4} sa \frac{4-x}{x} tako što ćete pomnožiti \frac{x-1}{x^{2}+4x+4} recipročnom vrijednošću od \frac{4-x}{x}.

$$\frac{x-2}{x^{2}+2x}-\frac{\left(x-1\right)x}{\left(x^{2}+4x+4\right)\left(4-x\right)}$$

Faktorirajte x^{2}+2x. Faktorirajte \left(x^{2}+4x+4\right)\left(4-x\right).

$$\frac{x-2}{x\left(x+2\right)}-\frac{\left(x-1\right)x}{\left(-x+4\right)\left(x+2\right)^{2}}$$

Da biste izvršili zbrajanje ili oduzimanje izraza, rastavite ih kako bi im nazivnici bili isti. Najmanji zajednički množilac brojeva x\left(x+2\right) i \left(-x+4\right)\left(x+2\right)^{2} je x\left(x-4\right)\left(x+2\right)^{2}. Pomnožite \frac{x-2}{x\left(x+2\right)} i \frac{\left(x-4\right)\left(x+2\right)}{\left(x-4\right)\left(x+2\right)}. Pomnožite \frac{\left(x-1\right)x}{\left(-x+4\right)\left(x+2\right)^{2}} i \frac{-x}{-x}.

$$\frac{\left(x-2\right)\left(x-4\right)\left(x+2\right)}{x\left(x-4\right)\left(x+2\right)^{2}}-\frac{\left(x-1\right)x\left(-1\right)x}{x\left(x-4\right)\left(x+2\right)^{2}}$$

Pošto \frac{\left(x-2\right)\left(x-4\right)\left(x+2\right)}{x\left(x-4\right)\left(x+2\right)^{2}} i \frac{\left(x-1\right)x\left(-1\right)x}{x\left(x-4\right)\left(x+2\right)^{2}} imaju isti imenilac, oduzmite ih tako što ćete oduzeti njihove brojioce.

$$\frac{\left(x-2\right)\left(x-4\right)\left(x+2\right)-\left(x-1\right)x\left(-1\right)x}{x\left(x-4\right)\left(x+2\right)^{2}}$$

Izvršite množenja u \left(x-2\right)\left(x-4\right)\left(x+2\right)-\left(x-1\right)x\left(-1\right)x.

$$\frac{x^{3}-2x^{2}-8x-2x^{2}+4x+16+x^{3}-x^{2}}{x\left(x-4\right)\left(x+2\right)^{2}}$$

Kombinirajte slične izraze u x^{3}-2x^{2}-8x-2x^{2}+4x+16+x^{3}-x^{2}.

$$\frac{2x^{3}-5x^{2}-4x+16}{x\left(x-4\right)\left(x+2\right)^{2}}$$

Proširite x\left(x-4\right)\left(x+2\right)^{2}.

$$\frac{2x^{3}-5x^{2}-4x+16}{x^{4}-12x^{2}-16x}$$

Proširi

$$\frac{-2x^{3}+5x^{2}+4x-16}{x\left(4-x\right)\left(x+2\right)^{2}}$$

Kratki koraci rješenja

Podijelite \frac{x-1}{x^{2}+4x+4} sa \frac{4-x}{x} tako što ćete pomnožiti \frac{x-1}{x^{2}+4x+4} recipročnom vrijednošću od \frac{4-x}{x}.

$$\frac{x-2}{x^{2}+2x}-\frac{\left(x-1\right)x}{\left(x^{2}+4x+4\right)\left(4-x\right)}$$

Faktorirajte x^{2}+2x. Faktorirajte \left(x^{2}+4x+4\right)\left(4-x\right).

$$\frac{x-2}{x\left(x+2\right)}-\frac{\left(x-1\right)x}{\left(-x+4\right)\left(x+2\right)^{2}}$$

Da biste izvršili zbrajanje ili oduzimanje izraza, rastavite ih kako bi im nazivnici bili isti. Najmanji zajednički množilac brojeva x\left(x+2\right) i \left(-x+4\right)\left(x+2\right)^{2} je x\left(x-4\right)\left(x+2\right)^{2}. Pomnožite \frac{x-2}{x\left(x+2\right)} i \frac{\left(x-4\right)\left(x+2\right)}{\left(x-4\right)\left(x+2\right)}. Pomnožite \frac{\left(x-1\right)x}{\left(-x+4\right)\left(x+2\right)^{2}} i \frac{-x}{-x}.

$$\frac{\left(x-2\right)\left(x-4\right)\left(x+2\right)}{x\left(x-4\right)\left(x+2\right)^{2}}-\frac{\left(x-1\right)x\left(-1\right)x}{x\left(x-4\right)\left(x+2\right)^{2}}$$

Pošto \frac{\left(x-2\right)\left(x-4\right)\left(x+2\right)}{x\left(x-4\right)\left(x+2\right)^{2}} i \frac{\left(x-1\right)x\left(-1\right)x}{x\left(x-4\right)\left(x+2\right)^{2}} imaju isti imenilac, oduzmite ih tako što ćete oduzeti njihove brojioce.

$$\frac{\left(x-2\right)\left(x-4\right)\left(x+2\right)-\left(x-1\right)x\left(-1\right)x}{x\left(x-4\right)\left(x+2\right)^{2}}$$

Izvršite množenja u \left(x-2\right)\left(x-4\right)\left(x+2\right)-\left(x-1\right)x\left(-1\right)x.

$$\frac{x^{3}-2x^{2}-8x-2x^{2}+4x+16+x^{3}-x^{2}}{x\left(x-4\right)\left(x+2\right)^{2}}$$

Kombinirajte slične izraze u x^{3}-2x^{2}-8x-2x^{2}+4x+16+x^{3}-x^{2}.

$$\frac{2x^{3}-5x^{2}-4x+16}{x\left(x-4\right)\left(x+2\right)^{2}}$$

Proširite x\left(x-4\right)\left(x+2\right)^{2}.

$$\frac{2x^{3}-5x^{2}-4x+16}{x^{4}-12x^{2}-16x}$$