Whakaoti i te 6x-36/x^2-36div12x^2-36x/x^2+9x-36=x+12/2x^2+12x

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

Whakaoti mō x (complex solution)

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

Ngā Hipanga Rongoā Poto

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2x\left(x+6\right).

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

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{6} ki te x+6.

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

Whakamahia te āhuatanga tuaritanga hei whakarea te \frac{1}{6}x+1 ki te 12+x ka whakakotahi i ngā kupu rite.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+\frac{1}{6}x^{2}+12 ki te \frac{6x-36}{x^{2}-36}.

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

Tuhia te 3\times \frac{6x-36}{x^{2}-36} hei hautanga kotahi.

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

Me whakarea te \frac{1}{6} ki te \frac{6x-36}{x^{2}-36} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

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

Tuhia te 12\times \frac{6x-36}{x^{2}-36} hei hautanga kotahi.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 6x-36.

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

Tuhia te \frac{18x-108}{x^{2}-36}x hei hautanga kotahi.

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

Me whakatauwehe ngā kīanga kāore anō i whakatauwehea i roto o \frac{6x-36}{6\left(x^{2}-36\right)}.

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

Me whakakore tahi te 6 i te taurunga me te tauraro.

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

Tuhia te \frac{x-6}{\left(x-6\right)\left(x+6\right)}x^{2} hei hautanga kotahi.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te 6x-36.

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

Tauwehea te x^{2}-36.

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

Tā te mea he rite te tauraro o \frac{\left(18x-108\right)x}{\left(x-6\right)\left(x+6\right)} me \frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

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

Mahia ngā whakarea i roto o \left(18x-108\right)x+\left(x-6\right)x^{2}.

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

Whakakotahitia ngā kupu rite i 18x^{2}-108x+x^{3}-6x^{2}.

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

Tauwehea te x^{2}-36.

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

Tā te mea he rite te tauraro o \frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)} me \frac{72x-432}{\left(x-6\right)\left(x+6\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

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

Whakakotahitia ngā kupu rite i 12x^{2}-108x+x^{3}+72x-432.

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

Whakaarohia te \left(x-6\right)\left(x+6\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 6.

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

Tangohia te x mai i ngā taha e rua.

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

Tauwehea te x^{2}-36.

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

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x ki te \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}.

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

Tā te mea he rite te tauraro o \frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)} me \frac{x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}, me tango rāua mā te tango i ō raua taurunga.

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

Mahia ngā whakarea i roto o 12x^{2}-36x+x^{3}-432-x\left(x-6\right)\left(x+6\right).

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

Whakakotahitia ngā kupu rite i 12x^{2}-36x+x^{3}-432-x^{3}-6x^{2}+6x^{2}+36x.

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

Tangohia te 12 mai i ngā taha e rua.

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

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 12 ki te \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}.

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

Tā te mea he rite te tauraro o \frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)} me \frac{12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}, me tango rāua mā te tango i ō raua taurunga.

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

Mahia ngā whakarea i roto o 12x^{2}-432-12\left(x-6\right)\left(x+6\right).

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

Whakakotahitia ngā kupu rite i 12x^{2}-432-12x^{2}-72x+72x+432.

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

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,6 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-6\right)\left(x+6\right).

$$0=0$$

He pono tēnei mō tētahi x ahakoa.

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

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,6,0.

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

Whakaoti mō x

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

Ngā Hipanga Rongoā Poto

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 2x\left(x+6\right).

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

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{1}{6} ki te x+6.

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

Whakamahia te āhuatanga tuaritanga hei whakarea te \frac{1}{6}x+1 ki te 12+x ka whakakotahi i ngā kupu rite.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+\frac{1}{6}x^{2}+12 ki te \frac{6x-36}{x^{2}-36}.

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

Tuhia te 3\times \frac{6x-36}{x^{2}-36} hei hautanga kotahi.

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

Me whakarea te \frac{1}{6} ki te \frac{6x-36}{x^{2}-36} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro.

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

Tuhia te 12\times \frac{6x-36}{x^{2}-36} hei hautanga kotahi.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te 6x-36.

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

Tuhia te \frac{18x-108}{x^{2}-36}x hei hautanga kotahi.

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

Me whakatauwehe ngā kīanga kāore anō i whakatauwehea i roto o \frac{6x-36}{6\left(x^{2}-36\right)}.

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

Me whakakore tahi te 6 i te taurunga me te tauraro.

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

Tuhia te \frac{x-6}{\left(x-6\right)\left(x+6\right)}x^{2} hei hautanga kotahi.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te 6x-36.

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

Tauwehea te x^{2}-36.

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

Tā te mea he rite te tauraro o \frac{\left(18x-108\right)x}{\left(x-6\right)\left(x+6\right)} me \frac{\left(x-6\right)x^{2}}{\left(x-6\right)\left(x+6\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

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

Mahia ngā whakarea i roto o \left(18x-108\right)x+\left(x-6\right)x^{2}.

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

Whakakotahitia ngā kupu rite i 18x^{2}-108x+x^{3}-6x^{2}.

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

Tauwehea te x^{2}-36.

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

Tā te mea he rite te tauraro o \frac{12x^{2}-108x+x^{3}}{\left(x-6\right)\left(x+6\right)} me \frac{72x-432}{\left(x-6\right)\left(x+6\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

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

Whakakotahitia ngā kupu rite i 12x^{2}-108x+x^{3}+72x-432.

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

Whakaarohia te \left(x-6\right)\left(x+6\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 6.

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

Tangohia te x mai i ngā taha e rua.

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

Tauwehea te x^{2}-36.

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

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x ki te \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}.

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

Tā te mea he rite te tauraro o \frac{12x^{2}-36x+x^{3}-432}{\left(x-6\right)\left(x+6\right)} me \frac{x\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}, me tango rāua mā te tango i ō raua taurunga.

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

Mahia ngā whakarea i roto o 12x^{2}-36x+x^{3}-432-x\left(x-6\right)\left(x+6\right).

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

Whakakotahitia ngā kupu rite i 12x^{2}-36x+x^{3}-432-x^{3}-6x^{2}+6x^{2}+36x.

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

Tangohia te 12 mai i ngā taha e rua.

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

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 12 ki te \frac{\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}.

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

Tā te mea he rite te tauraro o \frac{12x^{2}-432}{\left(x-6\right)\left(x+6\right)} me \frac{12\left(x-6\right)\left(x+6\right)}{\left(x-6\right)\left(x+6\right)}, me tango rāua mā te tango i ō raua taurunga.

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

Mahia ngā whakarea i roto o 12x^{2}-432-12\left(x-6\right)\left(x+6\right).

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

Whakakotahitia ngā kupu rite i 12x^{2}-432-12x^{2}-72x+72x+432.

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

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,6 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-6\right)\left(x+6\right).

$$0=0$$

He pono tēnei mō tētahi x ahakoa.

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

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,6,0.

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