Whakaoti i te 2/x^2-1-4/x^2-3x+2=1/3x^2+6x+3

$$\frac { 2 } { x ^ { 2 } - 1 } - \frac { 4 } { x ^ { 2 } - 3 x + 2 } = \frac { 1 } { 3 x ^ { 2 } + 6 x + 3 }$$

Whakaoti mō x

$$x = -\frac{13}{7} = -1\frac{6}{7} \approx -1.857142857$$

$$x=-2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,1,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(x-2\right)\left(x-1\right)\left(x+1\right)^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-1,x^{2}-3x+2,3x^{2}+6x+3.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x-2.

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

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-6 ki te x+1 ka whakakotahi i ngā kupu rite.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3x^{2}-3x-6 ki te 2.

$$6x^{2}-6x-12-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$6x^{2}-6x-12-3\left(x^{2}+2x+1\right)\times 4=\left(x-2\right)\left(x-1\right)$$

Whakareatia te 3 ki te 4, ka 12.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te x^{2}+2x+1.

$$6x^{2}-6x-12-\left(12x^{2}+24x+12\right)=\left(x-2\right)\left(x-1\right)$$

Hei kimi i te tauaro o 12x^{2}+24x+12, kimihia te tauaro o ia taurangi.

$$6x^{2}-6x-12-12x^{2}-24x-12=\left(x-2\right)\left(x-1\right)$$

Pahekotia te 6x^{2} me -12x^{2}, ka -6x^{2}.

$$-6x^{2}-6x-12-24x-12=\left(x-2\right)\left(x-1\right)$$

Pahekotia te -6x me -24x, ka -30x.

$$-6x^{2}-30x-12-12=\left(x-2\right)\left(x-1\right)$$

Tangohia te 12 i te -12, ka -24.

$$-6x^{2}-30x-24=\left(x-2\right)\left(x-1\right)$$

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

$$-6x^{2}-30x-24=x^{2}-3x+2$$

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

$$-6x^{2}-30x-24-x^{2}=-3x+2$$

Pahekotia te -6x^{2} me -x^{2}, ka -7x^{2}.

$$-7x^{2}-30x-24=-3x+2$$

Me tāpiri te 3x ki ngā taha e rua.

$$-7x^{2}-30x-24+3x=2$$

Pahekotia te -30x me 3x, ka -27x.

$$-7x^{2}-27x-24=2$$

Tangohia te 2 mai i ngā taha e rua.

$$-7x^{2}-27x-24-2=0$$

Tangohia te 2 i te -24, ka -26.

$$-7x^{2}-27x-26=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -7x^{2}+ax+bx-26. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-27 ab=-7\left(-26\right)=182$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 182.

$$-1,-182 -2,-91 -7,-26 -13,-14$$

Tātaihia te tapeke mō ia takirua.

$$-1-182=-183 -2-91=-93 -7-26=-33 -13-14=-27$$

Ko te otinga te takirua ka hoatu i te tapeke -27.

$$a=-13 b=-14$$

Tuhia anō te -7x^{2}-27x-26 hei \left(-7x^{2}-13x\right)+\left(-14x-26\right).

$$\left(-7x^{2}-13x\right)+\left(-14x-26\right)$$

Tauwehea te -x i te tuatahi me te -2 i te rōpū tuarua.

$$-x\left(7x+13\right)-2\left(7x+13\right)$$

Whakatauwehea atu te kīanga pātahi 7x+13 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7x+13\right)\left(-x-2\right)$$

Hei kimi otinga whārite, me whakaoti te 7x+13=0 me te -x-2=0.

$$x=-\frac{13}{7} x=-2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,1,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(x-2\right)\left(x-1\right)\left(x+1\right)^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-1,x^{2}-3x+2,3x^{2}+6x+3.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x-2.

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

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-6 ki te x+1 ka whakakotahi i ngā kupu rite.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3x^{2}-3x-6 ki te 2.

$$6x^{2}-6x-12-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$6x^{2}-6x-12-3\left(x^{2}+2x+1\right)\times 4=\left(x-2\right)\left(x-1\right)$$

Whakareatia te 3 ki te 4, ka 12.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te x^{2}+2x+1.

$$6x^{2}-6x-12-\left(12x^{2}+24x+12\right)=\left(x-2\right)\left(x-1\right)$$

Hei kimi i te tauaro o 12x^{2}+24x+12, kimihia te tauaro o ia taurangi.

$$6x^{2}-6x-12-12x^{2}-24x-12=\left(x-2\right)\left(x-1\right)$$

Pahekotia te 6x^{2} me -12x^{2}, ka -6x^{2}.

$$-6x^{2}-6x-12-24x-12=\left(x-2\right)\left(x-1\right)$$

Pahekotia te -6x me -24x, ka -30x.

$$-6x^{2}-30x-12-12=\left(x-2\right)\left(x-1\right)$$

Tangohia te 12 i te -12, ka -24.

$$-6x^{2}-30x-24=\left(x-2\right)\left(x-1\right)$$

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

$$-6x^{2}-30x-24=x^{2}-3x+2$$

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

$$-6x^{2}-30x-24-x^{2}=-3x+2$$

Pahekotia te -6x^{2} me -x^{2}, ka -7x^{2}.

$$-7x^{2}-30x-24=-3x+2$$

Me tāpiri te 3x ki ngā taha e rua.

$$-7x^{2}-30x-24+3x=2$$

Pahekotia te -30x me 3x, ka -27x.

$$-7x^{2}-27x-24=2$$

Tangohia te 2 mai i ngā taha e rua.

$$-7x^{2}-27x-24-2=0$$

Tangohia te 2 i te -24, ka -26.

$$-7x^{2}-27x-26=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -7 mō a, -27 mō b, me -26 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\left(-7\right)\left(-26\right)}}{2\left(-7\right)}$$

Pūrua -27.

$$x=\frac{-\left(-27\right)±\sqrt{729-4\left(-7\right)\left(-26\right)}}{2\left(-7\right)}$$

Whakareatia -4 ki te -7.

$$x=\frac{-\left(-27\right)±\sqrt{729+28\left(-26\right)}}{2\left(-7\right)}$$

Whakareatia 28 ki te -26.

$$x=\frac{-\left(-27\right)±\sqrt{729-728}}{2\left(-7\right)}$$

Tāpiri 729 ki te -728.

$$x=\frac{-\left(-27\right)±\sqrt{1}}{2\left(-7\right)}$$

Tuhia te pūtakerua o te 1.

$$x=\frac{-\left(-27\right)±1}{2\left(-7\right)}$$

Ko te tauaro o -27 ko 27.

$$x=\frac{27±1}{2\left(-7\right)}$$

Whakareatia 2 ki te -7.

$$x=\frac{27±1}{-14}$$

Nā, me whakaoti te whārite x=\frac{27±1}{-14} ina he tāpiri te ±. Tāpiri 27 ki te 1.

$$x=\frac{28}{-14}$$

Whakawehe 28 ki te -14.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{27±1}{-14} ina he tango te ±. Tango 1 mai i 27.

$$x=\frac{26}{-14}$$

Whakahekea te hautanga \frac{26}{-14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{13}{7}$$

Kua oti te whārite te whakatau.

$$x=-2 x=-\frac{13}{7}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,1,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(x-2\right)\left(x-1\right)\left(x+1\right)^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-1,x^{2}-3x+2,3x^{2}+6x+3.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x-2.

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

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-6 ki te x+1 ka whakakotahi i ngā kupu rite.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 3x^{2}-3x-6 ki te 2.

$$6x^{2}-6x-12-3\left(x+1\right)^{2}\times 4=\left(x-2\right)\left(x-1\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$6x^{2}-6x-12-3\left(x^{2}+2x+1\right)\times 4=\left(x-2\right)\left(x-1\right)$$

Whakareatia te 3 ki te 4, ka 12.

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

Whakamahia te āhuatanga tohatoha hei whakarea te 12 ki te x^{2}+2x+1.

$$6x^{2}-6x-12-\left(12x^{2}+24x+12\right)=\left(x-2\right)\left(x-1\right)$$

Hei kimi i te tauaro o 12x^{2}+24x+12, kimihia te tauaro o ia taurangi.

$$6x^{2}-6x-12-12x^{2}-24x-12=\left(x-2\right)\left(x-1\right)$$

Pahekotia te 6x^{2} me -12x^{2}, ka -6x^{2}.

$$-6x^{2}-6x-12-24x-12=\left(x-2\right)\left(x-1\right)$$

Pahekotia te -6x me -24x, ka -30x.

$$-6x^{2}-30x-12-12=\left(x-2\right)\left(x-1\right)$$

Tangohia te 12 i te -12, ka -24.

$$-6x^{2}-30x-24=\left(x-2\right)\left(x-1\right)$$

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

$$-6x^{2}-30x-24=x^{2}-3x+2$$

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

$$-6x^{2}-30x-24-x^{2}=-3x+2$$

Pahekotia te -6x^{2} me -x^{2}, ka -7x^{2}.

$$-7x^{2}-30x-24=-3x+2$$

Me tāpiri te 3x ki ngā taha e rua.

$$-7x^{2}-30x-24+3x=2$$

Pahekotia te -30x me 3x, ka -27x.

$$-7x^{2}-27x-24=2$$

Me tāpiri te 24 ki ngā taha e rua.

$$-7x^{2}-27x=2+24$$

Tāpirihia te 2 ki te 24, ka 26.

$$-7x^{2}-27x=26$$

Whakawehea ngā taha e rua ki te -7.

$$\frac{-7x^{2}-27x}{-7}=\frac{26}{-7}$$

Mā te whakawehe ki te -7 ka wetekia te whakareanga ki te -7.

$$x^{2}+\left(-\frac{27}{-7}\right)x=\frac{26}{-7}$$

Whakawehe -27 ki te -7.

$$x^{2}+\frac{27}{7}x=\frac{26}{-7}$$

Whakawehe 26 ki te -7.

$$x^{2}+\frac{27}{7}x=-\frac{26}{7}$$

Whakawehea te \frac{27}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{27}{14}. Nā, tāpiria te pūrua o te \frac{27}{14} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{27}{7}x+\left(\frac{27}{14}\right)^{2}=-\frac{26}{7}+\left(\frac{27}{14}\right)^{2}$$

Pūruatia \frac{27}{14} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{27}{7}x+\frac{729}{196}=-\frac{26}{7}+\frac{729}{196}$$

Tāpiri -\frac{26}{7} ki te \frac{729}{196} 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.

$$x^{2}+\frac{27}{7}x+\frac{729}{196}=\frac{1}{196}$$

Tauwehea x^{2}+\frac{27}{7}x+\frac{729}{196}. 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}.

$$\left(x+\frac{27}{14}\right)^{2}=\frac{1}{196}$$

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

$$\sqrt{\left(x+\frac{27}{14}\right)^{2}}=\sqrt{\frac{1}{196}}$$

Whakarūnātia.

$$x+\frac{27}{14}=\frac{1}{14} x+\frac{27}{14}=-\frac{1}{14}$$

Me tango \frac{27}{14} mai i ngā taha e rua o te whārite.

$$x=-\frac{13}{7} x=-2$$