Whakaoti i te 30/x^2+5x+6-x/x+2=2x+1/x+3

$$\frac { 30 } { x ^ { 2 } + 5 x + 6 } - \frac { x } { x + 2 } = \frac { 2 x + 1 } { x + 3 }$$

Whakaoti mō x

$$x = -\frac{14}{3} = -4\frac{2}{3} \approx -4.666666667$$

$$x=2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,-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+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+5x+6,x+2,x+3.

$$30-\left(x+3\right)x=\left(x+2\right)\left(2x+1\right)$$

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

$$30-\left(x^{2}+3x\right)=\left(x+2\right)\left(2x+1\right)$$

Hei kimi i te tauaro o x^{2}+3x, kimihia te tauaro o ia taurangi.

$$30-x^{2}-3x=\left(x+2\right)\left(2x+1\right)$$

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

$$30-x^{2}-3x=2x^{2}+5x+2$$

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

$$30-x^{2}-3x-2x^{2}=5x+2$$

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

$$30-3x^{2}-3x=5x+2$$

Tangohia te 5x mai i ngā taha e rua.

$$30-3x^{2}-3x-5x=2$$

Pahekotia te -3x me -5x, ka -8x.

$$30-3x^{2}-8x=2$$

Tangohia te 2 mai i ngā taha e rua.

$$30-3x^{2}-8x-2=0$$

Tangohia te 2 i te 30, ka 28.

$$28-3x^{2}-8x=0$$

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

$$-3x^{2}-8x+28=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -3x^{2}+ax+bx+28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=-3\times 28=-84$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -84.

$$1,-84 2,-42 3,-28 4,-21 6,-14 7,-12$$

Tātaihia te tapeke mō ia takirua.

$$1-84=-83 2-42=-40 3-28=-25 4-21=-17 6-14=-8 7-12=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=6 b=-14$$

Tuhia anō te -3x^{2}-8x+28 hei \left(-3x^{2}+6x\right)+\left(-14x+28\right).

$$\left(-3x^{2}+6x\right)+\left(-14x+28\right)$$

Tauwehea te 3x i te tuatahi me te 14 i te rōpū tuarua.

$$3x\left(-x+2\right)+14\left(-x+2\right)$$

Whakatauwehea atu te kīanga pātahi -x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+2\right)\left(3x+14\right)$$

Hei kimi otinga whārite, me whakaoti te -x+2=0 me te 3x+14=0.

$$x=2 x=-\frac{14}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,-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+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+5x+6,x+2,x+3.

$$30-\left(x+3\right)x=\left(x+2\right)\left(2x+1\right)$$

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

$$30-\left(x^{2}+3x\right)=\left(x+2\right)\left(2x+1\right)$$

Hei kimi i te tauaro o x^{2}+3x, kimihia te tauaro o ia taurangi.

$$30-x^{2}-3x=\left(x+2\right)\left(2x+1\right)$$

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

$$30-x^{2}-3x=2x^{2}+5x+2$$

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

$$30-x^{2}-3x-2x^{2}=5x+2$$

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

$$30-3x^{2}-3x=5x+2$$

Tangohia te 5x mai i ngā taha e rua.

$$30-3x^{2}-3x-5x=2$$

Pahekotia te -3x me -5x, ka -8x.

$$30-3x^{2}-8x=2$$

Tangohia te 2 mai i ngā taha e rua.

$$30-3x^{2}-8x-2=0$$

Tangohia te 2 i te 30, ka 28.

$$28-3x^{2}-8x=0$$

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.

$$-3x^{2}-8x+28=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, -8 mō b, me 28 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-3\right)\times 28}}{2\left(-3\right)}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-3\right)\times 28}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-\left(-8\right)±\sqrt{64+12\times 28}}{2\left(-3\right)}$$

Whakareatia 12 ki te 28.

$$x=\frac{-\left(-8\right)±\sqrt{64+336}}{2\left(-3\right)}$$

Tāpiri 64 ki te 336.

$$x=\frac{-\left(-8\right)±\sqrt{400}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 400.

$$x=\frac{-\left(-8\right)±20}{2\left(-3\right)}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±20}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{8±20}{-6}$$

Nā, me whakaoti te whārite x=\frac{8±20}{-6} ina he tāpiri te ±. Tāpiri 8 ki te 20.

$$x=\frac{28}{-6}$$

Whakahekea te hautanga \frac{28}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{14}{3}$$

Nā, me whakaoti te whārite x=\frac{8±20}{-6} ina he tango te ±. Tango 20 mai i 8.

$$x=-\frac{12}{-6}$$

Whakawehe -12 ki te -6.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=-\frac{14}{3} x=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,-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+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+5x+6,x+2,x+3.

$$30-\left(x+3\right)x=\left(x+2\right)\left(2x+1\right)$$

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

$$30-\left(x^{2}+3x\right)=\left(x+2\right)\left(2x+1\right)$$

Hei kimi i te tauaro o x^{2}+3x, kimihia te tauaro o ia taurangi.

$$30-x^{2}-3x=\left(x+2\right)\left(2x+1\right)$$

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

$$30-x^{2}-3x=2x^{2}+5x+2$$

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

$$30-x^{2}-3x-2x^{2}=5x+2$$

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

$$30-3x^{2}-3x=5x+2$$

Tangohia te 5x mai i ngā taha e rua.

$$30-3x^{2}-3x-5x=2$$

Pahekotia te -3x me -5x, ka -8x.

$$30-3x^{2}-8x=2$$

Tangohia te 30 mai i ngā taha e rua.

$$-3x^{2}-8x=2-30$$

Tangohia te 30 i te 2, ka -28.

$$-3x^{2}-8x=-28$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}-8x}{-3}=-\frac{28}{-3}$$

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

$$x^{2}+\left(-\frac{8}{-3}\right)x=-\frac{28}{-3}$$

Whakawehe -8 ki te -3.

$$x^{2}+\frac{8}{3}x=-\frac{28}{-3}$$

Whakawehe -28 ki te -3.

$$x^{2}+\frac{8}{3}x=\frac{28}{3}$$

Whakawehea te \frac{8}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{4}{3}. Nā, tāpiria te pūrua o te \frac{4}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{8}{3}x+\left(\frac{4}{3}\right)^{2}=\frac{28}{3}+\left(\frac{4}{3}\right)^{2}$$

Pūruatia \frac{4}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=\frac{28}{3}+\frac{16}{9}$$

Tāpiri \frac{28}{3} ki te \frac{16}{9} 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.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=\frac{100}{9}$$

Tauwehea te x^{2}+\frac{8}{3}x+\frac{16}{9}. 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}.

$$\left(x+\frac{4}{3}\right)^{2}=\frac{100}{9}$$

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

$$\sqrt{\left(x+\frac{4}{3}\right)^{2}}=\sqrt{\frac{100}{9}}$$

Whakarūnātia.

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

Me tango \frac{4}{3} mai i ngā taha e rua o te whārite.

$$x=2 x=-\frac{14}{3}$$