Riješi x^2+x/x^2-25+x^2-1/x^2+11x+30

$$\frac { x ^ { 2 } + x } { x ^ { 2 } - 25 } + \frac { x ^ { 2 } - 1 } { x ^ { 2 } + 11 x + 30 }$$

Izračunaj

$$\frac{\left(x+1\right)\left(2x^{2}+5\right)}{\left(x+6\right)\left(x^{2}-25\right)}$$

Kratki koraci za rješenje

Rastavite x^{2}-25 na faktore. Rastavite x^{2}+11x+30 na faktore.

$$\frac{x^{2}+x}{\left(x-5\right)\left(x+5\right)}+\frac{x^{2}-1}{\left(x+5\right)\left(x+6\right)}$$

Da biste zbrojili ili oduzeli izraze, proširite ih da bi imali iste nazivnike. Najmanji zajednički višekratnik brojeva \left(x-5\right)\left(x+5\right) i \left(x+5\right)\left(x+6\right) jest \left(x-5\right)\left(x+5\right)\left(x+6\right). Pomnožite \frac{x^{2}+x}{\left(x-5\right)\left(x+5\right)} i \frac{x+6}{x+6}. Pomnožite \frac{x^{2}-1}{\left(x+5\right)\left(x+6\right)} i \frac{x-5}{x-5}.

$$\frac{\left(x^{2}+x\right)\left(x+6\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}+\frac{\left(x^{2}-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Budući da \frac{\left(x^{2}+x\right)\left(x+6\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)} i \frac{\left(x^{2}-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)} imaju isti nazivnik, zbrojite ih zbrajanjem njihovih brojnika.

$$\frac{\left(x^{2}+x\right)\left(x+6\right)+\left(x^{2}-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Pomnožite izraz \left(x^{2}+x\right)\left(x+6\right)+\left(x^{2}-1\right)\left(x-5\right).

$$\frac{x^{3}+6x^{2}+x^{2}+6x+x^{3}-5x^{2}-x+5}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Kombinirajte slične izraze u x^{3}+6x^{2}+x^{2}+6x+x^{3}-5x^{2}-x+5.

$$\frac{2x^{3}+2x^{2}+5x+5}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Proširivanje broja \left(x-5\right)\left(x+5\right)\left(x+6\right).

$$\frac{2x^{3}+2x^{2}+5x+5}{x^{3}+6x^{2}-25x-150}$$

Proširi

$$\frac{2x^{3}+2x^{2}+5x+5}{\left(x+6\right)\left(x^{2}-25\right)}$$

Kratki koraci za rješenje

Rastavite x^{2}-25 na faktore. Rastavite x^{2}+11x+30 na faktore.

$$\frac{x^{2}+x}{\left(x-5\right)\left(x+5\right)}+\frac{x^{2}-1}{\left(x+5\right)\left(x+6\right)}$$

Da biste zbrojili ili oduzeli izraze, proširite ih da bi imali iste nazivnike. Najmanji zajednički višekratnik brojeva \left(x-5\right)\left(x+5\right) i \left(x+5\right)\left(x+6\right) jest \left(x-5\right)\left(x+5\right)\left(x+6\right). Pomnožite \frac{x^{2}+x}{\left(x-5\right)\left(x+5\right)} i \frac{x+6}{x+6}. Pomnožite \frac{x^{2}-1}{\left(x+5\right)\left(x+6\right)} i \frac{x-5}{x-5}.

$$\frac{\left(x^{2}+x\right)\left(x+6\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}+\frac{\left(x^{2}-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Budući da \frac{\left(x^{2}+x\right)\left(x+6\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)} i \frac{\left(x^{2}-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)} imaju isti nazivnik, zbrojite ih zbrajanjem njihovih brojnika.

$$\frac{\left(x^{2}+x\right)\left(x+6\right)+\left(x^{2}-1\right)\left(x-5\right)}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Pomnožite izraz \left(x^{2}+x\right)\left(x+6\right)+\left(x^{2}-1\right)\left(x-5\right).

$$\frac{x^{3}+6x^{2}+x^{2}+6x+x^{3}-5x^{2}-x+5}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Kombinirajte slične izraze u x^{3}+6x^{2}+x^{2}+6x+x^{3}-5x^{2}-x+5.

$$\frac{2x^{3}+2x^{2}+5x+5}{\left(x-5\right)\left(x+5\right)\left(x+6\right)}$$

Proširivanje broja \left(x-5\right)\left(x+5\right)\left(x+6\right).

$$\frac{2x^{3}+2x^{2}+5x+5}{x^{3}+6x^{2}-25x-150}$$