Whakaoti i te 21/x+1=16/x-2-6/x

$$\frac { 21 } { x + 1 } = \frac { 16 } { x - 2 } - \frac { 6 } { x }$$

Whakaoti mō x

$$x=-\frac{2}{11}\approx -0.181818182$$

$$x=6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,0,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-2\right)\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+1,x-2,x.

$$x\left(x-2\right)\times 21=x\left(x+1\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-2.

$$\left(x^{2}-2x\right)\times 21=x\left(x+1\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-2x ki te 21.

$$21x^{2}-42x=x\left(x+1\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

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

$$21x^{2}-42x=\left(x^{2}+x\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}+x ki te 16.

$$21x^{2}-42x=16x^{2}+16x-\left(x-2\right)\left(x+1\right)\times 6$$

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

$$21x^{2}-42x=16x^{2}+16x-\left(x^{2}-x-2\right)\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-x-2 ki te 6.

$$21x^{2}-42x=16x^{2}+16x-\left(6x^{2}-6x-12\right)$$

Hei kimi i te tauaro o 6x^{2}-6x-12, kimihia te tauaro o ia taurangi.

$$21x^{2}-42x=16x^{2}+16x-6x^{2}+6x+12$$

Pahekotia te 16x^{2} me -6x^{2}, ka 10x^{2}.

$$21x^{2}-42x=10x^{2}+16x+6x+12$$

Pahekotia te 16x me 6x, ka 22x.

$$21x^{2}-42x=10x^{2}+22x+12$$

Tangohia te 10x^{2} mai i ngā taha e rua.

$$21x^{2}-42x-10x^{2}=22x+12$$

Pahekotia te 21x^{2} me -10x^{2}, ka 11x^{2}.

$$11x^{2}-42x=22x+12$$

Tangohia te 22x mai i ngā taha e rua.

$$11x^{2}-42x-22x=12$$

Pahekotia te -42x me -22x, ka -64x.

$$11x^{2}-64x=12$$

Tangohia te 12 mai i ngā taha e rua.

$$11x^{2}-64x-12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 11 mō a, -64 mō b, me -12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\times 11\left(-12\right)}}{2\times 11}$$

Pūrua -64.

$$x=\frac{-\left(-64\right)±\sqrt{4096-4\times 11\left(-12\right)}}{2\times 11}$$

Whakareatia -4 ki te 11.

$$x=\frac{-\left(-64\right)±\sqrt{4096-44\left(-12\right)}}{2\times 11}$$

Whakareatia -44 ki te -12.

$$x=\frac{-\left(-64\right)±\sqrt{4096+528}}{2\times 11}$$

Tāpiri 4096 ki te 528.

$$x=\frac{-\left(-64\right)±\sqrt{4624}}{2\times 11}$$

Tuhia te pūtakerua o te 4624.

$$x=\frac{-\left(-64\right)±68}{2\times 11}$$

Ko te tauaro o -64 ko 64.

$$x=\frac{64±68}{2\times 11}$$

Whakareatia 2 ki te 11.

$$x=\frac{64±68}{22}$$

Nā, me whakaoti te whārite x=\frac{64±68}{22} ina he tāpiri te ±. Tāpiri 64 ki te 68.

$$x=\frac{132}{22}$$

Whakawehe 132 ki te 22.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{64±68}{22} ina he tango te ±. Tango 68 mai i 64.

$$x=-\frac{4}{22}$$

Whakahekea te hautanga \frac{-4}{22} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{2}{11}$$

Kua oti te whārite te whakatau.

$$x=6 x=-\frac{2}{11}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,0,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-2\right)\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+1,x-2,x.

$$x\left(x-2\right)\times 21=x\left(x+1\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-2.

$$\left(x^{2}-2x\right)\times 21=x\left(x+1\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-2x ki te 21.

$$21x^{2}-42x=x\left(x+1\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

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

$$21x^{2}-42x=\left(x^{2}+x\right)\times 16-\left(x-2\right)\left(x+1\right)\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}+x ki te 16.

$$21x^{2}-42x=16x^{2}+16x-\left(x-2\right)\left(x+1\right)\times 6$$

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

$$21x^{2}-42x=16x^{2}+16x-\left(x^{2}-x-2\right)\times 6$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-x-2 ki te 6.

$$21x^{2}-42x=16x^{2}+16x-\left(6x^{2}-6x-12\right)$$

Hei kimi i te tauaro o 6x^{2}-6x-12, kimihia te tauaro o ia taurangi.

$$21x^{2}-42x=16x^{2}+16x-6x^{2}+6x+12$$

Pahekotia te 16x^{2} me -6x^{2}, ka 10x^{2}.

$$21x^{2}-42x=10x^{2}+16x+6x+12$$

Pahekotia te 16x me 6x, ka 22x.

$$21x^{2}-42x=10x^{2}+22x+12$$

Tangohia te 10x^{2} mai i ngā taha e rua.

$$21x^{2}-42x-10x^{2}=22x+12$$

Pahekotia te 21x^{2} me -10x^{2}, ka 11x^{2}.

$$11x^{2}-42x=22x+12$$

Tangohia te 22x mai i ngā taha e rua.

$$11x^{2}-42x-22x=12$$

Pahekotia te -42x me -22x, ka -64x.

$$11x^{2}-64x=12$$

Whakawehea ngā taha e rua ki te 11.

$$\frac{11x^{2}-64x}{11}=\frac{12}{11}$$

Mā te whakawehe ki te 11 ka wetekia te whakareanga ki te 11.

$$x^{2}-\frac{64}{11}x=\frac{12}{11}$$

Whakawehea te -\frac{64}{11}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{32}{11}. Nā, tāpiria te pūrua o te -\frac{32}{11} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{64}{11}x+\left(-\frac{32}{11}\right)^{2}=\frac{12}{11}+\left(-\frac{32}{11}\right)^{2}$$

Pūruatia -\frac{32}{11} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{64}{11}x+\frac{1024}{121}=\frac{12}{11}+\frac{1024}{121}$$

Tāpiri \frac{12}{11} ki te \frac{1024}{121} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{64}{11}x+\frac{1024}{121}=\frac{1156}{121}$$

Tauwehea x^{2}-\frac{64}{11}x+\frac{1024}{121}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{32}{11}\right)^{2}=\frac{1156}{121}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-\frac{32}{11}\right)^{2}}=\sqrt{\frac{1156}{121}}$$

Whakarūnātia.

$$x-\frac{32}{11}=\frac{34}{11} x-\frac{32}{11}=-\frac{34}{11}$$

Me tāpiri \frac{32}{11} ki ngā taha e rua o te whārite.

$$x=6 x=-\frac{2}{11}$$