Riješi (x-2/x+1+5-x/x-2):[(1/x^2-x-2-1/x^2+3x+2)*(frac{x+1}{x}+frac{3-x^2}{x^2+x})]

$$( \frac { x - 2 } { x + 1 } + \frac { 5 - x } { x - 2 } ) : [ ( \frac { 1 } { x ^ { 2 } - x - 2 } - \frac { 1 } { x ^ { 2 } + 3 x + 2 } ) \cdot ( \frac { x + 1 } { x } + \frac { 3 - x ^ { 2 } } { x ^ { 2 } + x } ) ]$$

Procijeni

$$\frac{9x\left(x+1\right)}{8}$$

Kratki koraci rješenja

Da biste izvršili zbrajanje ili oduzimanje izraza, rastavite ih kako bi im nazivnici bili isti. Najmanji zajednički množilac brojeva x+1 i x-2 je \left(x-2\right)\left(x+1\right). Pomnožite \frac{x-2}{x+1} i \frac{x-2}{x-2}. Pomnožite \frac{5-x}{x-2} i \frac{x+1}{x+1}.

$$\frac{\frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}+\frac{\left(5-x\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Pošto \frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)} i \frac{\left(5-x\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} imaju isti imenilac, saberite ih tako što ćete sabrati njihove brojioce.

$$\frac{\frac{\left(x-2\right)\left(x-2\right)+\left(5-x\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Izvršite množenja u \left(x-2\right)\left(x-2\right)+\left(5-x\right)\left(x+1\right).

$$\frac{\frac{x^{2}-2x-2x+4+5x+5-x^{2}-x}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Kombinirajte slične izraze u x^{2}-2x-2x+4+5x+5-x^{2}-x.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Faktorirajte x^{2}-x-2. Faktorirajte x^{2}+3x+2.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{\left(x-2\right)\left(x+1\right)}-\frac{1}{\left(x+1\right)\left(x+2\right)}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

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

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x-2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

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

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{x+2-\left(x-2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

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

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{x+2-x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Kombinirajte slične izraze u x+2-x+2.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Faktorirajte x^{2}+x.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x\left(x+1\right)}\right)}$$

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

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{\left(x+1\right)\left(x+1\right)}{x\left(x+1\right)}+\frac{3-x^{2}}{x\left(x+1\right)}\right)}$$

Pošto \frac{\left(x+1\right)\left(x+1\right)}{x\left(x+1\right)} i \frac{3-x^{2}}{x\left(x+1\right)} imaju isti imenilac, saberite ih tako što ćete sabrati njihove brojioce.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\times \frac{\left(x+1\right)\left(x+1\right)+3-x^{2}}{x\left(x+1\right)}}$$

Izvršite množenja u \left(x+1\right)\left(x+1\right)+3-x^{2}.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\times \frac{x^{2}+x+1+x+3-x^{2}}{x\left(x+1\right)}}$$

Kombinirajte slične izraze u x^{2}+x+1+x+3-x^{2}.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\times \frac{2x+4}{x\left(x+1\right)}}$$

Pomnožite \frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} i \frac{2x+4}{x\left(x+1\right)} tako što ćete pomnožiti brojilac s brojiocem i imenilac s imeniocem.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4\left(2x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)}}$$

Podijelite \frac{9}{\left(x-2\right)\left(x+1\right)} sa \frac{4\left(2x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)} tako što ćete pomnožiti \frac{9}{\left(x-2\right)\left(x+1\right)} recipročnom vrijednošću od \frac{4\left(2x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)}.

$$\frac{9\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\times 4\left(2x+4\right)}$$

Otkaži \left(x-2\right)\left(x+1\right) u brojiocu i imeniocu.

$$\frac{9x\left(x+1\right)\left(x+2\right)}{4\left(2x+4\right)}$$

Faktorirajte izraze koji nisu već faktorirani.

$$\frac{9x\left(x+1\right)\left(x+2\right)}{2\times 4\left(x+2\right)}$$

Otkaži x+2 u brojiocu i imeniocu.

$$\frac{9x\left(x+1\right)}{2\times 4}$$

Razvijte izraz.

$$\frac{9x^{2}+9x}{8}$$

Proširi

$$\frac{9x^{2}+9x}{8}$$

Kratki koraci rješenja

Da biste izvršili zbrajanje ili oduzimanje izraza, rastavite ih kako bi im nazivnici bili isti. Najmanji zajednički množilac brojeva x+1 i x-2 je \left(x-2\right)\left(x+1\right). Pomnožite \frac{x-2}{x+1} i \frac{x-2}{x-2}. Pomnožite \frac{5-x}{x-2} i \frac{x+1}{x+1}.

$$\frac{\frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)}+\frac{\left(5-x\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Pošto \frac{\left(x-2\right)\left(x-2\right)}{\left(x-2\right)\left(x+1\right)} i \frac{\left(5-x\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)} imaju isti imenilac, saberite ih tako što ćete sabrati njihove brojioce.

$$\frac{\frac{\left(x-2\right)\left(x-2\right)+\left(5-x\right)\left(x+1\right)}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Izvršite množenja u \left(x-2\right)\left(x-2\right)+\left(5-x\right)\left(x+1\right).

$$\frac{\frac{x^{2}-2x-2x+4+5x+5-x^{2}-x}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Kombinirajte slične izraze u x^{2}-2x-2x+4+5x+5-x^{2}-x.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{x^{2}-x-2}-\frac{1}{x^{2}+3x+2}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Faktorirajte x^{2}-x-2. Faktorirajte x^{2}+3x+2.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{1}{\left(x-2\right)\left(x+1\right)}-\frac{1}{\left(x+1\right)\left(x+2\right)}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

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

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\left(\frac{x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}-\frac{x-2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\right)\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

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

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{x+2-\left(x-2\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

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

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{x+2-x+2}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Kombinirajte slične izraze u x+2-x+2.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x^{2}+x}\right)}$$

Faktorirajte x^{2}+x.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{x+1}{x}+\frac{3-x^{2}}{x\left(x+1\right)}\right)}$$

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

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\left(\frac{\left(x+1\right)\left(x+1\right)}{x\left(x+1\right)}+\frac{3-x^{2}}{x\left(x+1\right)}\right)}$$

Pošto \frac{\left(x+1\right)\left(x+1\right)}{x\left(x+1\right)} i \frac{3-x^{2}}{x\left(x+1\right)} imaju isti imenilac, saberite ih tako što ćete sabrati njihove brojioce.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\times \frac{\left(x+1\right)\left(x+1\right)+3-x^{2}}{x\left(x+1\right)}}$$

Izvršite množenja u \left(x+1\right)\left(x+1\right)+3-x^{2}.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\times \frac{x^{2}+x+1+x+3-x^{2}}{x\left(x+1\right)}}$$

Kombinirajte slične izraze u x^{2}+x+1+x+3-x^{2}.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)}\times \frac{2x+4}{x\left(x+1\right)}}$$

Pomnožite \frac{4}{\left(x-2\right)\left(x+1\right)\left(x+2\right)} i \frac{2x+4}{x\left(x+1\right)} tako što ćete pomnožiti brojilac s brojiocem i imenilac s imeniocem.

$$\frac{\frac{9}{\left(x-2\right)\left(x+1\right)}}{\frac{4\left(2x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)}}$$

Podijelite \frac{9}{\left(x-2\right)\left(x+1\right)} sa \frac{4\left(2x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)} tako što ćete pomnožiti \frac{9}{\left(x-2\right)\left(x+1\right)} recipročnom vrijednošću od \frac{4\left(2x+4\right)}{\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)}.

$$\frac{9\left(x-2\right)\left(x+1\right)\left(x+2\right)x\left(x+1\right)}{\left(x-2\right)\left(x+1\right)\times 4\left(2x+4\right)}$$

Otkaži \left(x-2\right)\left(x+1\right) u brojiocu i imeniocu.

$$\frac{9x\left(x+1\right)\left(x+2\right)}{4\left(2x+4\right)}$$

Faktorirajte izraze koji nisu već faktorirani.

$$\frac{9x\left(x+1\right)\left(x+2\right)}{2\times 4\left(x+2\right)}$$

Otkaži x+2 u brojiocu i imeniocu.

$$\frac{9x\left(x+1\right)}{2\times 4}$$

Razvijte izraz.

$$\frac{9x^{2}+9x}{8}$$