Whakaoti i te x/x^2-2x-5/3x^2-12=2/x

$$\frac { x } { x ^ { 2 } - 2 x } - \frac { 5 } { 3 x ^ { 2 } - 12 } = \frac { 2 } { x }$$

Whakaoti mō x

$$x = -\frac{8}{3} = -2\frac{2}{3} \approx -2.666666667$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,0,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3x\left(x-2\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-2x,3x^{2}-12,x.

$$\left(3x+6\right)x-x\times 5=\left(3x^{2}-12\right)\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+6 ki te x.

$$3x^{2}+6x-x\times 5=\left(3x^{2}-12\right)\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x^{2}-12 ki te 2.

$$3x^{2}+6x-x\times 5=6x^{2}-24$$

Tangohia te 6x^{2} mai i ngā taha e rua.

$$3x^{2}+6x-x\times 5-6x^{2}=-24$$

Pahekotia te 3x^{2} me -6x^{2}, ka -3x^{2}.

$$-3x^{2}+6x-x\times 5=-24$$

Me tāpiri te 24 ki ngā taha e rua.

$$-3x^{2}+6x-x\times 5+24=0$$

Whakareatia te -1 ki te 5, ka -5.

$$-3x^{2}+6x-5x+24=0$$

Pahekotia te 6x me -5x, ka x.

$$-3x^{2}+x+24=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -3x^{2}+ax+bx+24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=-3\times 24=-72$$

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 -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=9 b=-8$$

Tuhia anō te -3x^{2}+x+24 hei \left(-3x^{2}+9x\right)+\left(-8x+24\right).

$$\left(-3x^{2}+9x\right)+\left(-8x+24\right)$$

Tauwehea te 3x i te tuatahi me te 8 i te rōpū tuarua.

$$3x\left(-x+3\right)+8\left(-x+3\right)$$

Whakatauwehea atu te kīanga pātahi -x+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+3\right)\left(3x+8\right)$$

Hei kimi otinga whārite, me whakaoti te -x+3=0 me te 3x+8=0.

$$x=3 x=-\frac{8}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,0,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3x\left(x-2\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-2x,3x^{2}-12,x.

$$\left(3x+6\right)x-x\times 5=\left(3x^{2}-12\right)\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+6 ki te x.

$$3x^{2}+6x-x\times 5=\left(3x^{2}-12\right)\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x^{2}-12 ki te 2.

$$3x^{2}+6x-x\times 5=6x^{2}-24$$

Tangohia te 6x^{2} mai i ngā taha e rua.

$$3x^{2}+6x-x\times 5-6x^{2}=-24$$

Pahekotia te 3x^{2} me -6x^{2}, ka -3x^{2}.

$$-3x^{2}+6x-x\times 5=-24$$

Me tāpiri te 24 ki ngā taha e rua.

$$-3x^{2}+6x-x\times 5+24=0$$

Whakareatia te -1 ki te 5, ka -5.

$$-3x^{2}+6x-5x+24=0$$

Pahekotia te 6x me -5x, ka x.

$$-3x^{2}+x+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 1 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-3\right)\times 24}}{2\left(-3\right)}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\left(-3\right)\times 24}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-1±\sqrt{1+12\times 24}}{2\left(-3\right)}$$

Whakareatia 12 ki te 24.

$$x=\frac{-1±\sqrt{1+288}}{2\left(-3\right)}$$

Tāpiri 1 ki te 288.

$$x=\frac{-1±\sqrt{289}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 289.

$$x=\frac{-1±17}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-1±17}{-6}$$

Nā, me whakaoti te whārite x=\frac{-1±17}{-6} ina he tāpiri te ±. Tāpiri -1 ki te 17.

$$x=\frac{16}{-6}$$

Whakahekea te hautanga \frac{16}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{8}{3}$$

Nā, me whakaoti te whārite x=\frac{-1±17}{-6} ina he tango te ±. Tango 17 mai i -1.

$$x=-\frac{18}{-6}$$

Whakawehe -18 ki te -6.

$$x=3$$

Kua oti te whārite te whakatau.

$$x=-\frac{8}{3} x=3$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,0,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3x\left(x-2\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-2x,3x^{2}-12,x.

$$\left(3x+6\right)x-x\times 5=\left(3x^{2}-12\right)\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x+6 ki te x.

$$3x^{2}+6x-x\times 5=\left(3x^{2}-12\right)\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x^{2}-12 ki te 2.

$$3x^{2}+6x-x\times 5=6x^{2}-24$$

Tangohia te 6x^{2} mai i ngā taha e rua.

$$3x^{2}+6x-x\times 5-6x^{2}=-24$$

Pahekotia te 3x^{2} me -6x^{2}, ka -3x^{2}.

$$-3x^{2}+6x-x\times 5=-24$$

Whakareatia te -1 ki te 5, ka -5.

$$-3x^{2}+6x-5x=-24$$

Pahekotia te 6x me -5x, ka x.

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

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}+x}{-3}=-\frac{24}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$x^{2}+\frac{1}{-3}x=-\frac{24}{-3}$$

Whakawehe 1 ki te -3.

$$x^{2}-\frac{1}{3}x=-\frac{24}{-3}$$

Whakawehe -24 ki te -3.

$$x^{2}-\frac{1}{3}x=8$$

Whakawehea te -\frac{1}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{6}. Nā, tāpiria te pūrua o te -\frac{1}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{3}x+\left(-\frac{1}{6}\right)^{2}=8+\left(-\frac{1}{6}\right)^{2}$$

Pūruatia -\frac{1}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=8+\frac{1}{36}$$

Tāpiri 8 ki te \frac{1}{36}.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{289}{36}$$

Tauwehea te x^{2}-\frac{1}{3}x+\frac{1}{36}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{6}\right)^{2}=\frac{289}{36}$$

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

$$\sqrt{\left(x-\frac{1}{6}\right)^{2}}=\sqrt{\frac{289}{36}}$$

Whakarūnātia.

$$x-\frac{1}{6}=\frac{17}{6} x-\frac{1}{6}=-\frac{17}{6}$$

Me tāpiri \frac{1}{6} ki ngā taha e rua o te whārite.

$$x=3 x=-\frac{8}{3}$$