Whakaoti i te 7/x-3-10/x-2-6/x-1=0

$$\frac { 7 } { x - 3 } - \frac { 10 } { x - 2 } - \frac { 6 } { x - 1 } = 0$$

Whakaoti mō x

$$x=4$$

$$x = \frac{13}{9} = 1\frac{4}{9} \approx 1.444444444$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 1,2,3 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)\left(x-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x-2,x-1.

$$\left(x-2\right)\left(x-1\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$\left(x^{2}-3x+2\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-3x+2 ki te 7.

$$7x^{2}-21x+14-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$7x^{2}-21x+14-\left(x^{2}-4x+3\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-4x+3 ki te 10.

$$7x^{2}-21x+14-\left(10x^{2}-40x+30\right)-\left(x-3\right)\left(x-2\right)\times 6=0$$

Hei kimi i te tauaro o 10x^{2}-40x+30, kimihia te tauaro o ia taurangi.

$$7x^{2}-21x+14-10x^{2}+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Pahekotia te 7x^{2} me -10x^{2}, ka -3x^{2}.

$$-3x^{2}-21x+14+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Pahekotia te -21x me 40x, ka 19x.

$$-3x^{2}+19x+14-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Tangohia te 30 i te 14, ka -16.

$$-3x^{2}+19x-16-\left(x-3\right)\left(x-2\right)\times 6=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-3 ki te x-2 ka whakakotahi i ngā kupu rite.

$$-3x^{2}+19x-16-\left(x^{2}-5x+6\right)\times 6=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-5x+6 ki te 6.

$$-3x^{2}+19x-16-\left(6x^{2}-30x+36\right)=0$$

Hei kimi i te tauaro o 6x^{2}-30x+36, kimihia te tauaro o ia taurangi.

$$-3x^{2}+19x-16-6x^{2}+30x-36=0$$

Pahekotia te -3x^{2} me -6x^{2}, ka -9x^{2}.

$$-9x^{2}+19x-16+30x-36=0$$

Pahekotia te 19x me 30x, ka 49x.

$$-9x^{2}+49x-16-36=0$$

Tangohia te 36 i te -16, ka -52.

$$-9x^{2}+49x-52=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -9x^{2}+ax+bx-52. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=49 ab=-9\left(-52\right)=468$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 468.

$$1,468 2,234 3,156 4,117 6,78 9,52 12,39 13,36 18,26$$

Tātaihia te tapeke mō ia takirua.

$$1+468=469 2+234=236 3+156=159 4+117=121 6+78=84 9+52=61 12+39=51 13+36=49 18+26=44$$

Ko te otinga te takirua ka hoatu i te tapeke 49.

$$a=36 b=13$$

Tuhia anō te -9x^{2}+49x-52 hei \left(-9x^{2}+36x\right)+\left(13x-52\right).

$$\left(-9x^{2}+36x\right)+\left(13x-52\right)$$

Tauwehea te 9x i te tuatahi me te -13 i te rōpū tuarua.

$$9x\left(-x+4\right)-13\left(-x+4\right)$$

Whakatauwehea atu te kīanga pātahi -x+4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+4\right)\left(9x-13\right)$$

Hei kimi otinga whārite, me whakaoti te -x+4=0 me te 9x-13=0.

$$x=4 x=\frac{13}{9}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 1,2,3 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)\left(x-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x-2,x-1.

$$\left(x-2\right)\left(x-1\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$\left(x^{2}-3x+2\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-3x+2 ki te 7.

$$7x^{2}-21x+14-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$7x^{2}-21x+14-\left(x^{2}-4x+3\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-4x+3 ki te 10.

$$7x^{2}-21x+14-\left(10x^{2}-40x+30\right)-\left(x-3\right)\left(x-2\right)\times 6=0$$

Hei kimi i te tauaro o 10x^{2}-40x+30, kimihia te tauaro o ia taurangi.

$$7x^{2}-21x+14-10x^{2}+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Pahekotia te 7x^{2} me -10x^{2}, ka -3x^{2}.

$$-3x^{2}-21x+14+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Pahekotia te -21x me 40x, ka 19x.

$$-3x^{2}+19x+14-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Tangohia te 30 i te 14, ka -16.

$$-3x^{2}+19x-16-\left(x-3\right)\left(x-2\right)\times 6=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-3 ki te x-2 ka whakakotahi i ngā kupu rite.

$$-3x^{2}+19x-16-\left(x^{2}-5x+6\right)\times 6=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-5x+6 ki te 6.

$$-3x^{2}+19x-16-\left(6x^{2}-30x+36\right)=0$$

Hei kimi i te tauaro o 6x^{2}-30x+36, kimihia te tauaro o ia taurangi.

$$-3x^{2}+19x-16-6x^{2}+30x-36=0$$

Pahekotia te -3x^{2} me -6x^{2}, ka -9x^{2}.

$$-9x^{2}+19x-16+30x-36=0$$

Pahekotia te 19x me 30x, ka 49x.

$$-9x^{2}+49x-16-36=0$$

Tangohia te 36 i te -16, ka -52.

$$-9x^{2}+49x-52=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -9 mō a, 49 mō b, me -52 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-49±\sqrt{49^{2}-4\left(-9\right)\left(-52\right)}}{2\left(-9\right)}$$

Pūrua 49.

$$x=\frac{-49±\sqrt{2401-4\left(-9\right)\left(-52\right)}}{2\left(-9\right)}$$

Whakareatia -4 ki te -9.

$$x=\frac{-49±\sqrt{2401+36\left(-52\right)}}{2\left(-9\right)}$$

Whakareatia 36 ki te -52.

$$x=\frac{-49±\sqrt{2401-1872}}{2\left(-9\right)}$$

Tāpiri 2401 ki te -1872.

$$x=\frac{-49±\sqrt{529}}{2\left(-9\right)}$$

Tuhia te pūtakerua o te 529.

$$x=\frac{-49±23}{2\left(-9\right)}$$

Whakareatia 2 ki te -9.

$$x=\frac{-49±23}{-18}$$

Nā, me whakaoti te whārite x=\frac{-49±23}{-18} ina he tāpiri te ±. Tāpiri -49 ki te 23.

$$x=-\frac{26}{-18}$$

Whakahekea te hautanga \frac{-26}{-18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{13}{9}$$

Nā, me whakaoti te whārite x=\frac{-49±23}{-18} ina he tango te ±. Tango 23 mai i -49.

$$x=-\frac{72}{-18}$$

Whakawehe -72 ki te -18.

$$x=4$$

Kua oti te whārite te whakatau.

$$x=\frac{13}{9} x=4$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 1,2,3 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)\left(x-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x-2,x-1.

$$\left(x-2\right)\left(x-1\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$\left(x^{2}-3x+2\right)\times 7-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-3x+2 ki te 7.

$$7x^{2}-21x+14-\left(x-3\right)\left(x-1\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

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

$$7x^{2}-21x+14-\left(x^{2}-4x+3\right)\times 10-\left(x-3\right)\left(x-2\right)\times 6=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-4x+3 ki te 10.

$$7x^{2}-21x+14-\left(10x^{2}-40x+30\right)-\left(x-3\right)\left(x-2\right)\times 6=0$$

Hei kimi i te tauaro o 10x^{2}-40x+30, kimihia te tauaro o ia taurangi.

$$7x^{2}-21x+14-10x^{2}+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Pahekotia te 7x^{2} me -10x^{2}, ka -3x^{2}.

$$-3x^{2}-21x+14+40x-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Pahekotia te -21x me 40x, ka 19x.

$$-3x^{2}+19x+14-30-\left(x-3\right)\left(x-2\right)\times 6=0$$

Tangohia te 30 i te 14, ka -16.

$$-3x^{2}+19x-16-\left(x-3\right)\left(x-2\right)\times 6=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-3 ki te x-2 ka whakakotahi i ngā kupu rite.

$$-3x^{2}+19x-16-\left(x^{2}-5x+6\right)\times 6=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-5x+6 ki te 6.

$$-3x^{2}+19x-16-\left(6x^{2}-30x+36\right)=0$$

Hei kimi i te tauaro o 6x^{2}-30x+36, kimihia te tauaro o ia taurangi.

$$-3x^{2}+19x-16-6x^{2}+30x-36=0$$

Pahekotia te -3x^{2} me -6x^{2}, ka -9x^{2}.

$$-9x^{2}+19x-16+30x-36=0$$

Pahekotia te 19x me 30x, ka 49x.

$$-9x^{2}+49x-16-36=0$$

Tangohia te 36 i te -16, ka -52.

$$-9x^{2}+49x-52=0$$

Me tāpiri te 52 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-9x^{2}+49x=52$$

Whakawehea ngā taha e rua ki te -9.

$$\frac{-9x^{2}+49x}{-9}=\frac{52}{-9}$$

Mā te whakawehe ki te -9 ka wetekia te whakareanga ki te -9.

$$x^{2}+\frac{49}{-9}x=\frac{52}{-9}$$

Whakawehe 49 ki te -9.

$$x^{2}-\frac{49}{9}x=\frac{52}{-9}$$

Whakawehe 52 ki te -9.

$$x^{2}-\frac{49}{9}x=-\frac{52}{9}$$

Whakawehea te -\frac{49}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{49}{18}. Nā, tāpiria te pūrua o te -\frac{49}{18} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{49}{9}x+\left(-\frac{49}{18}\right)^{2}=-\frac{52}{9}+\left(-\frac{49}{18}\right)^{2}$$

Pūruatia -\frac{49}{18} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{49}{9}x+\frac{2401}{324}=-\frac{52}{9}+\frac{2401}{324}$$

Tāpiri -\frac{52}{9} ki te \frac{2401}{324} 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.

$$x^{2}-\frac{49}{9}x+\frac{2401}{324}=\frac{529}{324}$$

Tauwehea x^{2}-\frac{49}{9}x+\frac{2401}{324}. 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}.

$$\left(x-\frac{49}{18}\right)^{2}=\frac{529}{324}$$

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

$$\sqrt{\left(x-\frac{49}{18}\right)^{2}}=\sqrt{\frac{529}{324}}$$

Whakarūnātia.

$$x-\frac{49}{18}=\frac{23}{18} x-\frac{49}{18}=-\frac{23}{18}$$

Me tāpiri \frac{49}{18} ki ngā taha e rua o te whārite.

$$x=4 x=\frac{13}{9}$$