Resol alpha/beta+1+beta/alpha+1

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

Calcula

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

Passos de solució senzills

Per afegir o restar les expressions, amplieu-les perquè els denominadors coincideixin. El mínim comú múltiple de \beta +1 i \alpha +1 és \left(\alpha +1\right)\left(\beta +1\right). Multipliqueu \frac{\alpha }{\beta +1} per \frac{\alpha +1}{\alpha +1}. Multipliqueu \frac{\beta }{\alpha +1} per \frac{\beta +1}{\beta +1}.

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

Com que \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)} tenen el mateix denominador, afegiu-los mitjançant l'addició dels seus numeradors.

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

Feu les multiplicacions a \alpha \left(\alpha +1\right)+\beta \left(\beta +1\right).

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

Expandiu \left(\alpha +1\right)\left(\beta +1\right).

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

Diferencieu α

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