Riješi alpha/beta+1+beta/alpha+1

$$\frac { \alpha } { \beta + 1 } + \frac { \beta } { \alpha + 1 }$$

Izračunaj

$$\frac{\alpha ^{2}+\alpha +\beta ^{2}+\beta }{\left(\alpha +1\right)\left(\beta +1\right)}$$

Kratki koraci za rješenje

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

$$\frac{\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}+\frac{\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}$$

Budući da \frac{\alpha \left(\alpha +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} i \frac{\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)} imaju isti nazivnik, zbrojite ih zbrajanjem njihovih brojnika.

$$\frac{\alpha \left(\alpha +1\right)+\beta \left(\beta +1\right)}{\left(\alpha +1\right)\left(\beta +1\right)}$$

Pomnožite izraz \alpha \left(\alpha +1\right)+\beta \left(\beta +1\right).

$$\frac{\alpha ^{2}+\alpha +\beta ^{2}+\beta }{\left(\alpha +1\right)\left(\beta +1\right)}$$

Proširivanje broja \left(\alpha +1\right)\left(\beta +1\right).

$$\frac{\alpha ^{2}+\alpha +\beta ^{2}+\beta }{\alpha \beta +\alpha +\beta +1}$$

Diferenciraj u odnosu na α

$$\frac{\alpha ^{2}+2\alpha -\beta ^{2}-\beta +1}{\left(\beta +1\right)\left(\alpha +1\right)^{2}}$$