Rešite 6x-36/x^2-36div12x^2-36x/x^2+9x-36=x+12/2x^2+12x

$$\frac { 6 x - 36 } { x ^ { 2 } - 36 } \div \frac { 12 x ^ { 2 } - 36 x } { x ^ { 2 } + 9 x - 36 } = \frac { x + 12 } { 2 x ^ { 2 } + 12 x }$$

Rešitev za x (complex solution)

$$x\in \mathrm{C}\setminus -6,6,0,-12,3$$

Kratka navodila za rešitev

Spremenljivka x ne more biti enaka nobeni od vrednosti -6,0, ker deljenje z vrednostjo nič ni določeno. Pomnožite obe strani enačbe s/z 2x\left(x+6\right).

$$\frac{1}{6}\left(x+6\right)\left(12+x\right)\times \frac{6x-36}{x^{2}-36}=x+12$$

Uporabite distributivnost, da pomnožite \frac{1}{6} s/z x+6.

$$\left(\frac{1}{6}x+1\right)\left(12+x\right)\times \frac{6x-36}{x^{2}-36}=x+12$$

Uporabite lastnost distributivnosti za množenje \frac{1}{6}x+1 krat 12+x in kombiniranje pogojev podobnosti.

$$\left(3x+\frac{1}{6}x^{2}+12\right)\times \frac{6x-36}{x^{2}-36}=x+12$$

Uporabite distributivnost, da pomnožite 3x+\frac{1}{6}x^{2}+12 s/z \frac{6x-36}{x^{2}-36}.

$$3x\times \frac{6x-36}{x^{2}-36}+\frac{1}{6}x^{2}\times \frac{6x-36}{x^{2}-36}+12\times \frac{6x-36}{x^{2}-36}=x+12$$

Izrazite 3\times \frac{6x-36}{x^{2}-36} kot enojni ulomek.

$$\frac{3\left(6x-36\right)}{x^{2}-36}x+\frac{1}{6}x^{2}\times \frac{6x-36}{x^{2}-36}+12\times \frac{6x-36}{x^{2}-36}=x+12$$

Pomnožite \frac{1}{6} s/z \frac{6x-36}{x^{2}-36} tako, da pomnožite števec s števcem in imenovalec z imenovalcem.

$$\frac{3\left(6x-36\right)}{x^{2}-36}x+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+12\times \frac{6x-36}{x^{2}-36}=x+12$$

Izrazite 12\times \frac{6x-36}{x^{2}-36} kot enojni ulomek.

$$\frac{3\left(6x-36\right)}{x^{2}-36}x+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Uporabite distributivnost, da pomnožite 3 s/z 6x-36.

$$\frac{18x-108}{x^{2}-36}x+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Izrazite \frac{18x-108}{x^{2}-36}x kot enojni ulomek.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Faktorizirajte izraze, ki še niso faktorizirani v \frac{6x-36}{6\left(x^{2}-36\right)}.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{6\left(x-6\right)}{6\left(x-6\right)\left(x+6\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Okrajšaj 6 v števcu in imenovalcu.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{x-6}{\left(x-6\right)\left(x+6\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Izrazite \frac{x-6}{\left(x-6\right)\left(x+6\right)}x^{2} kot enojni ulomek.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Uporabite distributivnost, da pomnožite 12 s/z 6x-36.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

Faktorizirajte x^{2}-36.

$$\frac{\left(18x-108\right)x}{\left(x-6\right)\left(x+6\right)}+\frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

\frac{\left(18x-108\right)x}{\left(x-6\right)\left(x+6\right)} in \frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)} imata isti imenovalec, zato ju seštejte tako, da seštejete njuna števca.

$$\frac{\left(18x-108\right)x+\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

Izvedi množenje v \left(18x-108\right)x+\left(x-6\right)x^{2}.

$$\frac{18x^{2}-108x+x^{3}-6x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

Združite podobne člene v 18x^{2}-108x+x^{3}-6x^{2}.

$$\frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

Faktorizirajte x^{2}-36.

$$\frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{\left(x-6\right)\left(x+6\right)}=x+12$$

\frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)} in \frac{72x-432}{\left(x-6\right)\left(x+6\right)} imata isti imenovalec, zato ju seštejte tako, da seštejete njuna števca.

$$\frac{12x^{2}-108x+x^{3}+72x-432}{\left(x-6\right)\left(x+6\right)}=x+12$$

Združite podobne člene v 12x^{2}-108x+x^{3}+72x-432.

$$\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)}=x+12$$

Razmislite o \left(x-6\right)\left(x+6\right). Množenje je lahko preoblikovano v razliko kvadratov s pravilom: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Kvadrat števila 6.

$$\frac{12x^{2}-36x+x^{3}-432}{x^{2}-36}=x+12$$

Odštejte x na obeh straneh.

$$\frac{12x^{2}-36x+x^{3}-432}{x^{2}-36}-x=12$$

Faktorizirajte x^{2}-36.

$$\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)}-x=12$$

Če želite prišteti ali odšteti izraze, jih razširite na skupne imenovalce. Pomnožite x s/z \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}.

$$\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)}-\frac{x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=12$$

Ker \frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)} in \frac{x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} imata isti imenovalec, jih odštejte tako, da odštejete njihove števce.

$$\frac{12x^{2}-36x+x^{3}-432-x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=12$$

Izvedi množenje v 12x^{2}-36x+x^{3}-432-x\left(x-6\right)\left(x+6\right).

$$\frac{12x^{2}-36x+x^{3}-432-x^{3}-6x^{2}+6x^{2}+36x}{\left(x-6\right)\left(x+6\right)}=12$$

Združite podobne člene v 12x^{2}-36x+x^{3}-432-x^{3}-6x^{2}+6x^{2}+36x.

$$\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)}=12$$

Odštejte 12 na obeh straneh.

$$\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)}-12=0$$

Če želite prišteti ali odšteti izraze, jih razširite na skupne imenovalce. Pomnožite 12 s/z \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}.

$$\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)}-\frac{12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=0$$

Ker \frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)} in \frac{12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} imata isti imenovalec, jih odštejte tako, da odštejete njihove števce.

$$\frac{12x^{2}-432-12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=0$$

Izvedi množenje v 12x^{2}-432-12\left(x-6\right)\left(x+6\right).

$$\frac{12x^{2}-432-12x^{2}-72x+72x+432}{\left(x-6\right)\left(x+6\right)}=0$$

Združite podobne člene v 12x^{2}-432-12x^{2}-72x+72x+432.

$$\frac{0}{\left(x-6\right)\left(x+6\right)}=0$$

Spremenljivka x ne more biti enaka nobeni od vrednosti -6,6, ker deljenje z vrednostjo nič ni določeno. Pomnožite obe strani enačbe s/z \left(x-6\right)\left(x+6\right).

$$0=0$$

To je za vsak x »true«.

$$x\in \mathrm{C}$$

Spremenljivka x ne more biti enaka nobeni od vrednosti -6,6,0.

$$x\in \mathrm{C}\setminus -6,0,6$$

Rešitev za x

$$x\in \mathrm{R}\setminus 6,-6,0,3,-12$$

Kratka navodila za rešitev

Spremenljivka x ne more biti enaka nobeni od vrednosti -6,0, ker deljenje z vrednostjo nič ni določeno. Pomnožite obe strani enačbe s/z 2x\left(x+6\right).

$$\frac{1}{6}\left(x+6\right)\left(12+x\right)\times \frac{6x-36}{x^{2}-36}=x+12$$

Uporabite distributivnost, da pomnožite \frac{1}{6} s/z x+6.

$$\left(\frac{1}{6}x+1\right)\left(12+x\right)\times \frac{6x-36}{x^{2}-36}=x+12$$

Uporabite lastnost distributivnosti za množenje \frac{1}{6}x+1 krat 12+x in kombiniranje pogojev podobnosti.

$$\left(3x+\frac{1}{6}x^{2}+12\right)\times \frac{6x-36}{x^{2}-36}=x+12$$

Uporabite distributivnost, da pomnožite 3x+\frac{1}{6}x^{2}+12 s/z \frac{6x-36}{x^{2}-36}.

$$3x\times \frac{6x-36}{x^{2}-36}+\frac{1}{6}x^{2}\times \frac{6x-36}{x^{2}-36}+12\times \frac{6x-36}{x^{2}-36}=x+12$$

Izrazite 3\times \frac{6x-36}{x^{2}-36} kot enojni ulomek.

$$\frac{3\left(6x-36\right)}{x^{2}-36}x+\frac{1}{6}x^{2}\times \frac{6x-36}{x^{2}-36}+12\times \frac{6x-36}{x^{2}-36}=x+12$$

Pomnožite \frac{1}{6} s/z \frac{6x-36}{x^{2}-36} tako, da pomnožite števec s števcem in imenovalec z imenovalcem.

$$\frac{3\left(6x-36\right)}{x^{2}-36}x+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+12\times \frac{6x-36}{x^{2}-36}=x+12$$

Izrazite 12\times \frac{6x-36}{x^{2}-36} kot enojni ulomek.

$$\frac{3\left(6x-36\right)}{x^{2}-36}x+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Uporabite distributivnost, da pomnožite 3 s/z 6x-36.

$$\frac{18x-108}{x^{2}-36}x+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Izrazite \frac{18x-108}{x^{2}-36}x kot enojni ulomek.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{6x-36}{6\left(x^{2}-36\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Faktorizirajte izraze, ki še niso faktorizirani v \frac{6x-36}{6\left(x^{2}-36\right)}.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{6\left(x-6\right)}{6\left(x-6\right)\left(x+6\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Okrajšaj 6 v števcu in imenovalcu.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{x-6}{\left(x-6\right)\left(x+6\right)}x^{2}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Izrazite \frac{x-6}{\left(x-6\right)\left(x+6\right)}x^{2} kot enojni ulomek.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{12\left(6x-36\right)}{x^{2}-36}=x+12$$

Uporabite distributivnost, da pomnožite 12 s/z 6x-36.

$$\frac{\left(18x-108\right)x}{x^{2}-36}+\frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

Faktorizirajte x^{2}-36.

$$\frac{\left(18x-108\right)x}{\left(x-6\right)\left(x+6\right)}+\frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

\frac{\left(18x-108\right)x}{\left(x-6\right)\left(x+6\right)} in \frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)} imata isti imenovalec, zato ju seštejte tako, da seštejete njuna števca.

$$\frac{\left(18x-108\right)x+\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

Izvedi množenje v \left(18x-108\right)x+\left(x-6\right)x^{2}.

$$\frac{18x^{2}-108x+x^{3}-6x^{2}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

Združite podobne člene v 18x^{2}-108x+x^{3}-6x^{2}.

$$\frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{x^{2}-36}=x+12$$

Faktorizirajte x^{2}-36.

$$\frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)}+\frac{72x-432}{\left(x-6\right)\left(x+6\right)}=x+12$$

\frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)} in \frac{72x-432}{\left(x-6\right)\left(x+6\right)} imata isti imenovalec, zato ju seštejte tako, da seštejete njuna števca.

$$\frac{12x^{2}-108x+x^{3}+72x-432}{\left(x-6\right)\left(x+6\right)}=x+12$$

Združite podobne člene v 12x^{2}-108x+x^{3}+72x-432.

$$\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)}=x+12$$

Razmislite o \left(x-6\right)\left(x+6\right). Množenje je lahko preoblikovano v razliko kvadratov s pravilom: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Kvadrat števila 6.

$$\frac{12x^{2}-36x+x^{3}-432}{x^{2}-36}=x+12$$

Odštejte x na obeh straneh.

$$\frac{12x^{2}-36x+x^{3}-432}{x^{2}-36}-x=12$$

Faktorizirajte x^{2}-36.

$$\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)}-x=12$$

Če želite prišteti ali odšteti izraze, jih razširite na skupne imenovalce. Pomnožite x s/z \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}.

$$\frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)}-\frac{x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=12$$

Ker \frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)} in \frac{x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} imata isti imenovalec, jih odštejte tako, da odštejete njihove števce.

$$\frac{12x^{2}-36x+x^{3}-432-x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=12$$

Izvedi množenje v 12x^{2}-36x+x^{3}-432-x\left(x-6\right)\left(x+6\right).

$$\frac{12x^{2}-36x+x^{3}-432-x^{3}-6x^{2}+6x^{2}+36x}{\left(x-6\right)\left(x+6\right)}=12$$

Združite podobne člene v 12x^{2}-36x+x^{3}-432-x^{3}-6x^{2}+6x^{2}+36x.

$$\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)}=12$$

Odštejte 12 na obeh straneh.

$$\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)}-12=0$$

Če želite prišteti ali odšteti izraze, jih razširite na skupne imenovalce. Pomnožite 12 s/z \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}.

$$\frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)}-\frac{12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=0$$

Ker \frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)} in \frac{12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)} imata isti imenovalec, jih odštejte tako, da odštejete njihove števce.

$$\frac{12x^{2}-432-12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}=0$$

Izvedi množenje v 12x^{2}-432-12\left(x-6\right)\left(x+6\right).

$$\frac{12x^{2}-432-12x^{2}-72x+72x+432}{\left(x-6\right)\left(x+6\right)}=0$$

Združite podobne člene v 12x^{2}-432-12x^{2}-72x+72x+432.

$$\frac{0}{\left(x-6\right)\left(x+6\right)}=0$$

Spremenljivka x ne more biti enaka nobeni od vrednosti -6,6, ker deljenje z vrednostjo nič ni določeno. Pomnožite obe strani enačbe s/z \left(x-6\right)\left(x+6\right).

$$0=0$$

To je za vsak x »true«.

$$x\in \mathrm{R}$$

Spremenljivka x ne more biti enaka nobeni od vrednosti -6,6,0.

$$x\in \mathrm{R}\setminus -6,0,6$$