Whakaoti i te 5/x^2-4+x/x-2=4

$$\frac { 5 } { x ^ { 2 } - 4 } + \frac { x } { x - 2 } = 4$$

Whakaoti mō x

$$x = -\frac{7}{3} = -2\frac{1}{3} \approx -2.333333333$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,2 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+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-4,x-2.

$$5+\left(x+2\right)x=4\left(x-2\right)\left(x+2\right)$$

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

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

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x-2.

$$5+x^{2}+2x=\left(4x-8\right)\left(x+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 4x-8 ki te x+2 ka whakakotahi i ngā kupu rite.

$$5+x^{2}+2x=4x^{2}-16$$

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

$$5+x^{2}+2x-4x^{2}=-16$$

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

$$5-3x^{2}+2x=-16$$

Me tāpiri te 16 ki ngā taha e rua.

$$5-3x^{2}+2x+16=0$$

Tāpirihia te 5 ki te 16, ka 21.

$$21-3x^{2}+2x=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-3x^{2}+2x+21=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -3x^{2}+ax+bx+21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=-3\times 21=-63$$

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 -63.

$$-1,63 -3,21 -7,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+63=62 -3+21=18 -7+9=2$$

Ko te otinga te takirua ka hoatu i te tapeke 2.

$$a=9 b=-7$$

Tuhia anō te -3x^{2}+2x+21 hei \left(-3x^{2}+9x\right)+\left(-7x+21\right).

$$\left(-3x^{2}+9x\right)+\left(-7x+21\right)$$

Tauwehea te 3x i te tuatahi me te 7 i te rōpū tuarua.

$$3x\left(-x+3\right)+7\left(-x+3\right)$$

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

$$\left(-x+3\right)\left(3x+7\right)$$

Hei kimi otinga whārite, me whakaoti te -x+3=0 me te 3x+7=0.

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,2 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+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-4,x-2.

$$5+\left(x+2\right)x=4\left(x-2\right)\left(x+2\right)$$

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

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

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x-2.

$$5+x^{2}+2x=\left(4x-8\right)\left(x+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 4x-8 ki te x+2 ka whakakotahi i ngā kupu rite.

$$5+x^{2}+2x=4x^{2}-16$$

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

$$5+x^{2}+2x-4x^{2}=-16$$

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

$$5-3x^{2}+2x=-16$$

Me tāpiri te 16 ki ngā taha e rua.

$$5-3x^{2}+2x+16=0$$

Tāpirihia te 5 ki te 16, ka 21.

$$21-3x^{2}+2x=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$-3x^{2}+2x+21=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 2 mō b, me 21 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-3\right)\times 21}}{2\left(-3\right)}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\left(-3\right)\times 21}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-2±\sqrt{4+12\times 21}}{2\left(-3\right)}$$

Whakareatia 12 ki te 21.

$$x=\frac{-2±\sqrt{4+252}}{2\left(-3\right)}$$

Tāpiri 4 ki te 252.

$$x=\frac{-2±\sqrt{256}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-2±16}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-2±16}{-6}$$

Nā, me whakaoti te whārite x=\frac{-2±16}{-6} ina he tāpiri te ±. Tāpiri -2 ki te 16.

$$x=\frac{14}{-6}$$

Whakahekea te hautanga \frac{14}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Nā, me whakaoti te whārite x=\frac{-2±16}{-6} ina he tango te ±. Tango 16 mai i -2.

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

Whakawehe -18 ki te -6.

$$x=3$$

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,2 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+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-4,x-2.

$$5+\left(x+2\right)x=4\left(x-2\right)\left(x+2\right)$$

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

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

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x-2.

$$5+x^{2}+2x=\left(4x-8\right)\left(x+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 4x-8 ki te x+2 ka whakakotahi i ngā kupu rite.

$$5+x^{2}+2x=4x^{2}-16$$

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

$$5+x^{2}+2x-4x^{2}=-16$$

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

$$5-3x^{2}+2x=-16$$

Tangohia te 5 mai i ngā taha e rua.

$$-3x^{2}+2x=-16-5$$

Tangohia te 5 i te -16, ka -21.

$$-3x^{2}+2x=-21$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}+2x}{-3}=-\frac{21}{-3}$$

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

$$x^{2}+\frac{2}{-3}x=-\frac{21}{-3}$$

Whakawehe 2 ki te -3.

$$x^{2}-\frac{2}{3}x=-\frac{21}{-3}$$

Whakawehe -21 ki te -3.

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

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

$$x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=7+\left(-\frac{1}{3}\right)^{2}$$

Pūruatia -\frac{1}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=7+\frac{1}{9}$$

Tāpiri 7 ki te \frac{1}{9}.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{64}{9}$$

Tauwehea x^{2}-\frac{2}{3}x+\frac{1}{9}. 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}.

$$\left(x-\frac{1}{3}\right)^{2}=\frac{64}{9}$$

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

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{\frac{64}{9}}$$

Whakarūnātia.

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

Me tāpiri \frac{1}{3} ki ngā taha e rua o te whārite.

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