Løs 2/x^2+7x+10+3/x^2-25

$$\frac { 2 } { x ^ { 2 } + 7 x + 10 } + \frac { 3 } { x ^ { 2 } - 25 }$$

Evaluer

$$\frac{5x-4}{\left(x+2\right)\left(x^{2}-25\right)}$$

Korte løsningstrin

Faktoriser x^{2}+7x+10. Faktoriser x^{2}-25.

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

For tilføje eller fratrække udtryk skal du udvide dem for at gøre nævneren ens. Mindste fælles multiplum for \left(x+2\right)\left(x+5\right) og \left(x-5\right)\left(x+5\right) er \left(x-5\right)\left(x+2\right)\left(x+5\right). Multiplicer \frac{2}{\left(x+2\right)\left(x+5\right)} gange \frac{x-5}{x-5}. Multiplicer \frac{3}{\left(x-5\right)\left(x+5\right)} gange \frac{x+2}{x+2}.

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

Da \frac{2\left(x-5\right)}{\left(x-5\right)\left(x+2\right)\left(x+5\right)} og \frac{3\left(x+2\right)}{\left(x-5\right)\left(x+2\right)\left(x+5\right)} har den samme fællesnævner, skal du addere dem ved at tilføje deres tællere.

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

Lav multiplikationerne i 2\left(x-5\right)+3\left(x+2\right).

$$\frac{2x-10+3x+6}{\left(x-5\right)\left(x+2\right)\left(x+5\right)}$$

Kombiner ens led i 2x-10+3x+6.

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

Udvid \left(x-5\right)\left(x+2\right)\left(x+5\right).

$$\frac{5x-4}{x^{3}+2x^{2}-25x-50}$$

Differentier w.r.t. x

$$\frac{2\left(-5x^{3}+x^{2}+8x-175\right)}{\left(\left(x+2\right)\left(x^{2}-25\right)\right)^{2}}$$