Whakaoti i te 5/x-3-x-1/x-2=7

$$\frac { 5 } { x - 3 } - \frac { x - 1 } { x - 2 } = 7$$

Whakaoti mō x

$$x = \frac{\sqrt{11} + 11}{4} \approx 3.579156198$$

$$x = \frac{11 - \sqrt{11}}{4} \approx 1.920843802$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$\left(x-2\right)\times 5-\left(x-3\right)\left(x-1\right)=7\left(x-3\right)\left(x-2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 5.

$$5x-10-\left(x-3\right)\left(x-1\right)=7\left(x-3\right)\left(x-2\right)$$

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

$$5x-10-\left(x^{2}-4x+3\right)=7\left(x-3\right)\left(x-2\right)$$

Hei kimi i te tauaro o x^{2}-4x+3, kimihia te tauaro o ia taurangi.

$$5x-10-x^{2}+4x-3=7\left(x-3\right)\left(x-2\right)$$

Pahekotia te 5x me 4x, ka 9x.

$$9x-10-x^{2}-3=7\left(x-3\right)\left(x-2\right)$$

Tangohia te 3 i te -10, ka -13.

$$9x-13-x^{2}=7\left(x-3\right)\left(x-2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x-3.

$$9x-13-x^{2}=\left(7x-21\right)\left(x-2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 7x-21 ki te x-2 ka whakakotahi i ngā kupu rite.

$$9x-13-x^{2}=7x^{2}-35x+42$$

Tangohia te 7x^{2} mai i ngā taha e rua.

$$9x-13-x^{2}-7x^{2}=-35x+42$$

Pahekotia te -x^{2} me -7x^{2}, ka -8x^{2}.

$$9x-13-8x^{2}=-35x+42$$

Me tāpiri te 35x ki ngā taha e rua.

$$9x-13-8x^{2}+35x=42$$

Pahekotia te 9x me 35x, ka 44x.

$$44x-13-8x^{2}=42$$

Tangohia te 42 mai i ngā taha e rua.

$$44x-13-8x^{2}-42=0$$

Tangohia te 42 i te -13, ka -55.

$$44x-55-8x^{2}=0$$

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.

$$-8x^{2}+44x-55=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -8 mō a, 44 mō b, me -55 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-44±\sqrt{44^{2}-4\left(-8\right)\left(-55\right)}}{2\left(-8\right)}$$

Pūrua 44.

$$x=\frac{-44±\sqrt{1936-4\left(-8\right)\left(-55\right)}}{2\left(-8\right)}$$

Whakareatia -4 ki te -8.

$$x=\frac{-44±\sqrt{1936+32\left(-55\right)}}{2\left(-8\right)}$$

Whakareatia 32 ki te -55.

$$x=\frac{-44±\sqrt{1936-1760}}{2\left(-8\right)}$$

Tāpiri 1936 ki te -1760.

$$x=\frac{-44±\sqrt{176}}{2\left(-8\right)}$$

Tuhia te pūtakerua o te 176.

$$x=\frac{-44±4\sqrt{11}}{2\left(-8\right)}$$

Whakareatia 2 ki te -8.

$$x=\frac{-44±4\sqrt{11}}{-16}$$

Nā, me whakaoti te whārite x=\frac{-44±4\sqrt{11}}{-16} ina he tāpiri te ±. Tāpiri -44 ki te 4\sqrt{11}.

$$x=\frac{4\sqrt{11}-44}{-16}$$

Whakawehe -44+4\sqrt{11} ki te -16.

$$x=\frac{11-\sqrt{11}}{4}$$

Nā, me whakaoti te whārite x=\frac{-44±4\sqrt{11}}{-16} ina he tango te ±. Tango 4\sqrt{11} mai i -44.

$$x=\frac{-4\sqrt{11}-44}{-16}$$

Whakawehe -44-4\sqrt{11} ki te -16.

$$x=\frac{\sqrt{11}+11}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{11-\sqrt{11}}{4} x=\frac{\sqrt{11}+11}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$\left(x-2\right)\times 5-\left(x-3\right)\left(x-1\right)=7\left(x-3\right)\left(x-2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 5.

$$5x-10-\left(x-3\right)\left(x-1\right)=7\left(x-3\right)\left(x-2\right)$$

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

$$5x-10-\left(x^{2}-4x+3\right)=7\left(x-3\right)\left(x-2\right)$$

Hei kimi i te tauaro o x^{2}-4x+3, kimihia te tauaro o ia taurangi.

$$5x-10-x^{2}+4x-3=7\left(x-3\right)\left(x-2\right)$$

Pahekotia te 5x me 4x, ka 9x.

$$9x-10-x^{2}-3=7\left(x-3\right)\left(x-2\right)$$

Tangohia te 3 i te -10, ka -13.

$$9x-13-x^{2}=7\left(x-3\right)\left(x-2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 7 ki te x-3.

$$9x-13-x^{2}=\left(7x-21\right)\left(x-2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 7x-21 ki te x-2 ka whakakotahi i ngā kupu rite.

$$9x-13-x^{2}=7x^{2}-35x+42$$

Tangohia te 7x^{2} mai i ngā taha e rua.

$$9x-13-x^{2}-7x^{2}=-35x+42$$

Pahekotia te -x^{2} me -7x^{2}, ka -8x^{2}.

$$9x-13-8x^{2}=-35x+42$$

Me tāpiri te 35x ki ngā taha e rua.

$$9x-13-8x^{2}+35x=42$$

Pahekotia te 9x me 35x, ka 44x.

$$44x-13-8x^{2}=42$$

Me tāpiri te 13 ki ngā taha e rua.

$$44x-8x^{2}=42+13$$

Tāpirihia te 42 ki te 13, ka 55.

$$44x-8x^{2}=55$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$-8x^{2}+44x=55$$

Whakawehea ngā taha e rua ki te -8.

$$\frac{-8x^{2}+44x}{-8}=\frac{55}{-8}$$

Mā te whakawehe ki te -8 ka wetekia te whakareanga ki te -8.

$$x^{2}+\frac{44}{-8}x=\frac{55}{-8}$$

Whakahekea te hautanga \frac{44}{-8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{11}{2}x=\frac{55}{-8}$$

Whakawehe 55 ki te -8.

$$x^{2}-\frac{11}{2}x=-\frac{55}{8}$$

Whakawehea te -\frac{11}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{4}. Nā, tāpiria te pūrua o te -\frac{11}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{11}{2}x+\left(-\frac{11}{4}\right)^{2}=-\frac{55}{8}+\left(-\frac{11}{4}\right)^{2}$$

Pūruatia -\frac{11}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=-\frac{55}{8}+\frac{121}{16}$$

Tāpiri -\frac{55}{8} ki te \frac{121}{16} 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.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{11}{16}$$

Tauwehea x^{2}-\frac{11}{2}x+\frac{121}{16}. 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}.

$$\left(x-\frac{11}{4}\right)^{2}=\frac{11}{16}$$

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

$$\sqrt{\left(x-\frac{11}{4}\right)^{2}}=\sqrt{\frac{11}{16}}$$

Whakarūnātia.

$$x-\frac{11}{4}=\frac{\sqrt{11}}{4} x-\frac{11}{4}=-\frac{\sqrt{11}}{4}$$

Me tāpiri \frac{11}{4} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{11}+11}{4} x=\frac{11-\sqrt{11}}{4}$$