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

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

Whakaoti mō x

$$x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

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

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

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

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

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

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

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

Pahekotia te 2x^{2} me x^{2}, ka 3x^{2}.

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

Pahekotia te 6x me -3x, ka 3x.

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

Tāpirihia te 4 ki te 2, ka 6.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-1 ki te 4.

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

Tangohia te 4x^{2} mai i ngā taha e rua.

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

Pahekotia te 3x^{2} me -4x^{2}, ka -x^{2}.

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

Me tāpiri te 4 ki ngā taha e rua.

$$-x^{2}+3x+6+4=0$$

Tāpirihia te 6 ki te 4, ka 10.

$$-x^{2}+3x+10=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=-10=-10$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -10.

$$-1,10 -2,5$$

Tātaihia te tapeke mō ia takirua.

$$-1+10=9 -2+5=3$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=5 b=-2$$

Tuhia anō te -x^{2}+3x+10 hei \left(-x^{2}+5x\right)+\left(-2x+10\right).

$$\left(-x^{2}+5x\right)+\left(-2x+10\right)$$

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

$$-x\left(x-5\right)-2\left(x-5\right)$$

Whakatauwehea atu te kīanga pātahi x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-5\right)\left(-x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te -x-2=0.

$$x=5 x=-2$$

Tē taea kia ōrite te tāupe x ki -2.

$$x=5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

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

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

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

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

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

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

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

Pahekotia te 2x^{2} me x^{2}, ka 3x^{2}.

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

Pahekotia te 6x me -3x, ka 3x.

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

Tāpirihia te 4 ki te 2, ka 6.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-1 ki te 4.

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

Tangohia te 4x^{2} mai i ngā taha e rua.

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

Pahekotia te 3x^{2} me -4x^{2}, ka -x^{2}.

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

Me tāpiri te 4 ki ngā taha e rua.

$$-x^{2}+3x+6+4=0$$

Tāpirihia te 6 ki te 4, ka 10.

$$-x^{2}+3x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 3 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-3±\sqrt{9+4\times 10}}{2\left(-1\right)}$$

Whakareatia 4 ki te 10.

$$x=\frac{-3±\sqrt{9+40}}{2\left(-1\right)}$$

Tāpiri 9 ki te 40.

$$x=\frac{-3±\sqrt{49}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-3±7}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-3±7}{-2}$$

Nā, me whakaoti te whārite x=\frac{-3±7}{-2} ina he tāpiri te ±. Tāpiri -3 ki te 7.

$$x=\frac{4}{-2}$$

Whakawehe 4 ki te -2.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-3±7}{-2} ina he tango te ±. Tango 7 mai i -3.

$$x=-\frac{10}{-2}$$

Whakawehe -10 ki te -2.

$$x=5$$

Kua oti te whārite te whakatau.

$$x=-2 x=5$$

Tē taea kia ōrite te tāupe x ki -2.

$$x=5$$

Ngā Upane mō te Whakaoti i te Pūrua

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

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

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

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

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

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

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

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

Pahekotia te 2x^{2} me x^{2}, ka 3x^{2}.

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

Pahekotia te 6x me -3x, ka 3x.

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

Tāpirihia te 4 ki te 2, ka 6.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-1 ki te 4.

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

Tangohia te 4x^{2} mai i ngā taha e rua.

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

Pahekotia te 3x^{2} me -4x^{2}, ka -x^{2}.

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

Tangohia te 6 mai i ngā taha e rua.

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

Tangohia te 6 i te -4, ka -10.

$$-x^{2}+3x=-10$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+3x}{-1}=-\frac{10}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{3}{-1}x=-\frac{10}{-1}$$

Whakawehe 3 ki te -1.

$$x^{2}-3x=-\frac{10}{-1}$$

Whakawehe -10 ki te -1.

$$x^{2}-3x=10$$

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=10+\frac{9}{4}$$

Tāpiri 10 ki te \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{49}{4}$$

Tauwehea x^{2}-3x+\frac{9}{4}. 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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$x-\frac{3}{2}=\frac{7}{2} x-\frac{3}{2}=-\frac{7}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=5 x=-2$$

Tē taea kia ōrite te tāupe x ki -2.

$$x=5$$