Whakaoti i te 2x-1/16-x^2*x^2+3x-4/2x^2-3x+1=1/4-x

$$\frac { 2 x - 1 } { 16 - x ^ { 2 } } \cdot \frac { x ^ { 2 } + 3 x - 4 } { 2 x ^ { 2 } - 3 x + 1 } = \frac { 1 } { 4 - x }$$

Whakaoti mō x (complex solution)

$$x\in \mathrm{C}\setminus 4,-4,\frac{1}{2},1$$

Ngā Hipanga Rongoā Poto

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -4,\frac{1}{2},1,4 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-4\right)\left(x-1\right)\left(2x-1\right)\left(x+4\right), arā, te tauraro pātahi he tino iti rawa te kitea o 16-x^{2},2x^{2}-3x+1,4-x.

$$\left(-1+3x-2x^{2}\right)\left(2x-1\right)\times \frac{x^{2}+3x-4}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Tuhia te \left(-1+3x-2x^{2}\right)\times \frac{x^{2}+3x-4}{2x^{2}-3x+1} hei hautanga kotahi.

$$\frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1}\left(2x-1\right)=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1} ki te 2x-1.

$$2\times \frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1}x-\frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -1+3x-2x^{2} ki te x^{2}+3x-4 ka whakakotahi i ngā kupu rite.

$$2\times \frac{16x^{2}-15x+4-3x^{3}-2x^{4}}{2x^{2}-3x+1}x-\frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Tuhia te 2\times \frac{16x^{2}-15x+4-3x^{3}-2x^{4}}{2x^{2}-3x+1} hei hautanga kotahi.

$$\frac{2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)}{2x^{2}-3x+1}x-\frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Tuhia te \frac{2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)}{2x^{2}-3x+1}x hei hautanga kotahi.

$$\frac{2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)x}{2x^{2}-3x+1}-\frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -1+3x-2x^{2} ki te x^{2}+3x-4 ka whakakotahi i ngā kupu rite.

$$\frac{2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)x}{2x^{2}-3x+1}-\frac{16x^{2}-15x+4-3x^{3}-2x^{4}}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Tā te mea he rite te tauraro o \frac{2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)x}{2x^{2}-3x+1} me \frac{16x^{2}-15x+4-3x^{3}-2x^{4}}{2x^{2}-3x+1}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)x-\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Mahia ngā whakarea i roto o 2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)x-\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right).

$$\frac{32x^{3}-30x^{2}+8x-6x^{4}-4x^{5}-16x^{2}+15x-4+3x^{3}+2x^{4}}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Whakakotahitia ngā kupu rite i 32x^{3}-30x^{2}+8x-6x^{4}-4x^{5}-16x^{2}+15x-4+3x^{3}+2x^{4}.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1 ki te -1+x.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{2x^{2}-3x+1}=\left(1-x\right)\left(-1+2x\right)\left(4+x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 1-x ki te -1+2x ka whakakotahi i ngā kupu rite.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{2x^{2}-3x+1}=\left(-1+3x-2x^{2}\right)\left(4+x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -1+3x-2x^{2} ki te 4+x ka whakakotahi i ngā kupu rite.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{2x^{2}-3x+1}=-4+11x-5x^{2}-2x^{3}$$

Tangohia te -4 mai i ngā taha e rua.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{2x^{2}-3x+1}-\left(-4\right)=11x-5x^{2}-2x^{3}$$

Ko te tauaro o -4 ko 4.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{2x^{2}-3x+1}+4=11x-5x^{2}-2x^{3}$$

Tauwehea te 2x^{2}-3x+1.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{\left(x-1\right)\left(2x-1\right)}+4=11x-5x^{2}-2x^{3}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 4 ki te \frac{\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{\left(x-1\right)\left(2x-1\right)}+\frac{4\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=11x-5x^{2}-2x^{3}$$

Tā te mea he rite te tauraro o \frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{\left(x-1\right)\left(2x-1\right)} me \frac{4\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4+4\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=11x-5x^{2}-2x^{3}$$

Mahia ngā whakarea i roto o 35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4+4\left(x-1\right)\left(2x-1\right).

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4+8x^{2}-4x-8x+4}{\left(x-1\right)\left(2x-1\right)}=11x-5x^{2}-2x^{3}$$

Whakakotahitia ngā kupu rite i 35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4+8x^{2}-4x-8x+4.

$$\frac{35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}=11x-5x^{2}-2x^{3}$$

Tangohia te 11x mai i ngā taha e rua.

$$\frac{35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}-11x=-5x^{2}-2x^{3}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia -11x ki te \frac{\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}.

$$\frac{35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}+\frac{-11x\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=-5x^{2}-2x^{3}$$

Tā te mea he rite te tauraro o \frac{35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)} me \frac{-11x\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}-11x\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=-5x^{2}-2x^{3}$$

Mahia ngā whakarea i roto o 35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}-11x\left(x-1\right)\left(2x-1\right).

$$\frac{35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}-22x^{3}+11x^{2}+22x^{2}-11x}{\left(x-1\right)\left(2x-1\right)}=-5x^{2}-2x^{3}$$

Whakakotahitia ngā kupu rite i 35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}-22x^{3}+11x^{2}+22x^{2}-11x.

$$\frac{13x^{3}-5x^{2}-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}=-5x^{2}-2x^{3}$$

Me tāpiri te 5x^{2} ki ngā taha e rua.

$$\frac{13x^{3}-5x^{2}-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}+5x^{2}=-2x^{3}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 5x^{2} ki te \frac{\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}.

$$\frac{13x^{3}-5x^{2}-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}+\frac{5x^{2}\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=-2x^{3}$$

Tā te mea he rite te tauraro o \frac{13x^{3}-5x^{2}-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)} me \frac{5x^{2}\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{13x^{3}-5x^{2}-4x^{4}-4x^{5}+5x^{2}\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=-2x^{3}$$

Mahia ngā whakarea i roto o 13x^{3}-5x^{2}-4x^{4}-4x^{5}+5x^{2}\left(x-1\right)\left(2x-1\right).

$$\frac{13x^{3}-5x^{2}-4x^{4}-4x^{5}+10x^{4}-5x^{3}-10x^{3}+5x^{2}}{\left(x-1\right)\left(2x-1\right)}=-2x^{3}$$

Whakakotahitia ngā kupu rite i 13x^{3}-5x^{2}-4x^{4}-4x^{5}+10x^{4}-5x^{3}-10x^{3}+5x^{2}.

$$\frac{-2x^{3}+6x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}=-2x^{3}$$

Me tāpiri te 2x^{3} ki ngā taha e rua.

$$\frac{-2x^{3}+6x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}+2x^{3}=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 2x^{3} ki te \frac{\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}.

$$\frac{-2x^{3}+6x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}+\frac{2x^{3}\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=0$$

Tā te mea he rite te tauraro o \frac{-2x^{3}+6x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)} me \frac{2x^{3}\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{-2x^{3}+6x^{4}-4x^{5}+2x^{3}\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=0$$

Mahia ngā whakarea i roto o -2x^{3}+6x^{4}-4x^{5}+2x^{3}\left(x-1\right)\left(2x-1\right).

$$\frac{-2x^{3}+6x^{4}-4x^{5}+4x^{5}-2x^{4}-4x^{4}+2x^{3}}{\left(x-1\right)\left(2x-1\right)}=0$$

Whakakotahitia ngā kupu rite i -2x^{3}+6x^{4}-4x^{5}+4x^{5}-2x^{4}-4x^{4}+2x^{3}.

$$\frac{0}{\left(x-1\right)\left(2x-1\right)}=0$$

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara \frac{1}{2},1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-1\right)\left(2x-1\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 \frac{1}{2},1,-4,4.

$$x\in \mathrm{C}\setminus -4,\frac{1}{2},1,4$$

Whakaoti mō x

$$x\in \mathrm{R}\setminus 4,-4,1,\frac{1}{2}$$

Ngā Hipanga Rongoā Poto

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -4,\frac{1}{2},1,4 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-4\right)\left(x-1\right)\left(2x-1\right)\left(x+4\right), arā, te tauraro pātahi he tino iti rawa te kitea o 16-x^{2},2x^{2}-3x+1,4-x.

$$\left(-1+3x-2x^{2}\right)\left(2x-1\right)\times \frac{x^{2}+3x-4}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Tuhia te \left(-1+3x-2x^{2}\right)\times \frac{x^{2}+3x-4}{2x^{2}-3x+1} hei hautanga kotahi.

$$\frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1}\left(2x-1\right)=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1} ki te 2x-1.

$$2\times \frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1}x-\frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -1+3x-2x^{2} ki te x^{2}+3x-4 ka whakakotahi i ngā kupu rite.

$$2\times \frac{16x^{2}-15x+4-3x^{3}-2x^{4}}{2x^{2}-3x+1}x-\frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Tuhia te 2\times \frac{16x^{2}-15x+4-3x^{3}-2x^{4}}{2x^{2}-3x+1} hei hautanga kotahi.

$$\frac{2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)}{2x^{2}-3x+1}x-\frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Tuhia te \frac{2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)}{2x^{2}-3x+1}x hei hautanga kotahi.

$$\frac{2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)x}{2x^{2}-3x+1}-\frac{\left(-1+3x-2x^{2}\right)\left(x^{2}+3x-4\right)}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -1+3x-2x^{2} ki te x^{2}+3x-4 ka whakakotahi i ngā kupu rite.

$$\frac{2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)x}{2x^{2}-3x+1}-\frac{16x^{2}-15x+4-3x^{3}-2x^{4}}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Tā te mea he rite te tauraro o \frac{2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)x}{2x^{2}-3x+1} me \frac{16x^{2}-15x+4-3x^{3}-2x^{4}}{2x^{2}-3x+1}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)x-\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Mahia ngā whakarea i roto o 2\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right)x-\left(16x^{2}-15x+4-3x^{3}-2x^{4}\right).

$$\frac{32x^{3}-30x^{2}+8x-6x^{4}-4x^{5}-16x^{2}+15x-4+3x^{3}+2x^{4}}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Whakakotahitia ngā kupu rite i 32x^{3}-30x^{2}+8x-6x^{4}-4x^{5}-16x^{2}+15x-4+3x^{3}+2x^{4}.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{2x^{2}-3x+1}=-\left(-1+x\right)\left(-1+2x\right)\left(4+x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -1 ki te -1+x.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{2x^{2}-3x+1}=\left(1-x\right)\left(-1+2x\right)\left(4+x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 1-x ki te -1+2x ka whakakotahi i ngā kupu rite.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{2x^{2}-3x+1}=\left(-1+3x-2x^{2}\right)\left(4+x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -1+3x-2x^{2} ki te 4+x ka whakakotahi i ngā kupu rite.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{2x^{2}-3x+1}=-4+11x-5x^{2}-2x^{3}$$

Tangohia te -4 mai i ngā taha e rua.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{2x^{2}-3x+1}-\left(-4\right)=11x-5x^{2}-2x^{3}$$

Ko te tauaro o -4 ko 4.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{2x^{2}-3x+1}+4=11x-5x^{2}-2x^{3}$$

Tauwehea te 2x^{2}-3x+1.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{\left(x-1\right)\left(2x-1\right)}+4=11x-5x^{2}-2x^{3}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 4 ki te \frac{\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{\left(x-1\right)\left(2x-1\right)}+\frac{4\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=11x-5x^{2}-2x^{3}$$

Tā te mea he rite te tauraro o \frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4}{\left(x-1\right)\left(2x-1\right)} me \frac{4\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4+4\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=11x-5x^{2}-2x^{3}$$

Mahia ngā whakarea i roto o 35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4+4\left(x-1\right)\left(2x-1\right).

$$\frac{35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4+8x^{2}-4x-8x+4}{\left(x-1\right)\left(2x-1\right)}=11x-5x^{2}-2x^{3}$$

Whakakotahitia ngā kupu rite i 35x^{3}-46x^{2}+23x-4x^{4}-4x^{5}-4+8x^{2}-4x-8x+4.

$$\frac{35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}=11x-5x^{2}-2x^{3}$$

Tangohia te 11x mai i ngā taha e rua.

$$\frac{35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}-11x=-5x^{2}-2x^{3}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia -11x ki te \frac{\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}.

$$\frac{35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}+\frac{-11x\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=-5x^{2}-2x^{3}$$

Tā te mea he rite te tauraro o \frac{35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)} me \frac{-11x\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}-11x\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=-5x^{2}-2x^{3}$$

Mahia ngā whakarea i roto o 35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}-11x\left(x-1\right)\left(2x-1\right).

$$\frac{35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}-22x^{3}+11x^{2}+22x^{2}-11x}{\left(x-1\right)\left(2x-1\right)}=-5x^{2}-2x^{3}$$

Whakakotahitia ngā kupu rite i 35x^{3}-38x^{2}+11x-4x^{4}-4x^{5}-22x^{3}+11x^{2}+22x^{2}-11x.

$$\frac{13x^{3}-5x^{2}-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}=-5x^{2}-2x^{3}$$

Me tāpiri te 5x^{2} ki ngā taha e rua.

$$\frac{13x^{3}-5x^{2}-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}+5x^{2}=-2x^{3}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 5x^{2} ki te \frac{\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}.

$$\frac{13x^{3}-5x^{2}-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}+\frac{5x^{2}\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=-2x^{3}$$

Tā te mea he rite te tauraro o \frac{13x^{3}-5x^{2}-4x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)} me \frac{5x^{2}\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{13x^{3}-5x^{2}-4x^{4}-4x^{5}+5x^{2}\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=-2x^{3}$$

Mahia ngā whakarea i roto o 13x^{3}-5x^{2}-4x^{4}-4x^{5}+5x^{2}\left(x-1\right)\left(2x-1\right).

$$\frac{13x^{3}-5x^{2}-4x^{4}-4x^{5}+10x^{4}-5x^{3}-10x^{3}+5x^{2}}{\left(x-1\right)\left(2x-1\right)}=-2x^{3}$$

Whakakotahitia ngā kupu rite i 13x^{3}-5x^{2}-4x^{4}-4x^{5}+10x^{4}-5x^{3}-10x^{3}+5x^{2}.

$$\frac{-2x^{3}+6x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}=-2x^{3}$$

Me tāpiri te 2x^{3} ki ngā taha e rua.

$$\frac{-2x^{3}+6x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}+2x^{3}=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia 2x^{3} ki te \frac{\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}.

$$\frac{-2x^{3}+6x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)}+\frac{2x^{3}\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=0$$

Tā te mea he rite te tauraro o \frac{-2x^{3}+6x^{4}-4x^{5}}{\left(x-1\right)\left(2x-1\right)} me \frac{2x^{3}\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{-2x^{3}+6x^{4}-4x^{5}+2x^{3}\left(x-1\right)\left(2x-1\right)}{\left(x-1\right)\left(2x-1\right)}=0$$

Mahia ngā whakarea i roto o -2x^{3}+6x^{4}-4x^{5}+2x^{3}\left(x-1\right)\left(2x-1\right).

$$\frac{-2x^{3}+6x^{4}-4x^{5}+4x^{5}-2x^{4}-4x^{4}+2x^{3}}{\left(x-1\right)\left(2x-1\right)}=0$$

Whakakotahitia ngā kupu rite i -2x^{3}+6x^{4}-4x^{5}+4x^{5}-2x^{4}-4x^{4}+2x^{3}.

$$\frac{0}{\left(x-1\right)\left(2x-1\right)}=0$$

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara \frac{1}{2},1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-1\right)\left(2x-1\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 \frac{1}{2},1,-4,4.

$$x\in \mathrm{R}\setminus -4,\frac{1}{2},1,4$$