Whakaoti i te (x-2)/(x-3)-(x-4)/(x-5)=10/3

$$\frac { ( x - 2 ) } { ( x - 3 ) } - \frac { ( x - 4 ) } { ( x - 5 ) } = \frac { 10 } { 3 }$$

Whakaoti mō x

$$x=\frac{\sqrt{10}}{5}+4\approx 4.632455532$$

$$x=-\frac{\sqrt{10}}{5}+4\approx 3.367544468$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 3,5 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(x-5\right)\left(x-3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x-5,3.

$$\left(3x-15\right)\left(x-2\right)-\left(3x-9\right)\left(x-4\right)=10\left(x-5\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-15 ki te x-2 ka whakakotahi i ngā kupu rite.

$$3x^{2}-21x+30-\left(3x-9\right)\left(x-4\right)=10\left(x-5\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-9 ki te x-4 ka whakakotahi i ngā kupu rite.

$$3x^{2}-21x+30-\left(3x^{2}-21x+36\right)=10\left(x-5\right)\left(x-3\right)$$

Hei kimi i te tauaro o 3x^{2}-21x+36, kimihia te tauaro o ia taurangi.

$$3x^{2}-21x+30-3x^{2}+21x-36=10\left(x-5\right)\left(x-3\right)$$

Pahekotia te 3x^{2} me -3x^{2}, ka 0.

$$-21x+30+21x-36=10\left(x-5\right)\left(x-3\right)$$

Pahekotia te -21x me 21x, ka 0.

$$30-36=10\left(x-5\right)\left(x-3\right)$$

Tangohia te 36 i te 30, ka -6.

$$-6=10\left(x-5\right)\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10 ki te x-5.

$$-6=\left(10x-50\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 10x-50 ki te x-3 ka whakakotahi i ngā kupu rite.

$$-6=10x^{2}-80x+150$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$10x^{2}-80x+150=-6$$

Me tāpiri te 6 ki ngā taha e rua.

$$10x^{2}-80x+150+6=0$$

Tāpirihia te 150 ki te 6, ka 156.

$$10x^{2}-80x+156=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 10 mō a, -80 mō b, me 156 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 10\times 156}}{2\times 10}$$

Pūrua -80.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\times 10\times 156}}{2\times 10}$$

Whakareatia -4 ki te 10.

$$x=\frac{-\left(-80\right)±\sqrt{6400-40\times 156}}{2\times 10}$$

Whakareatia -40 ki te 156.

$$x=\frac{-\left(-80\right)±\sqrt{6400-6240}}{2\times 10}$$

Tāpiri 6400 ki te -6240.

$$x=\frac{-\left(-80\right)±\sqrt{160}}{2\times 10}$$

Tuhia te pūtakerua o te 160.

$$x=\frac{-\left(-80\right)±4\sqrt{10}}{2\times 10}$$

Ko te tauaro o -80 ko 80.

$$x=\frac{80±4\sqrt{10}}{2\times 10}$$

Whakareatia 2 ki te 10.

$$x=\frac{80±4\sqrt{10}}{20}$$

Nā, me whakaoti te whārite x=\frac{80±4\sqrt{10}}{20} ina he tāpiri te ±. Tāpiri 80 ki te 4\sqrt{10}.

$$x=\frac{4\sqrt{10}+80}{20}$$

Whakawehe 80+4\sqrt{10} ki te 20.

$$x=\frac{\sqrt{10}}{5}+4$$

Nā, me whakaoti te whārite x=\frac{80±4\sqrt{10}}{20} ina he tango te ±. Tango 4\sqrt{10} mai i 80.

$$x=\frac{80-4\sqrt{10}}{20}$$

Whakawehe 80-4\sqrt{10} ki te 20.

$$x=-\frac{\sqrt{10}}{5}+4$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{10}}{5}+4 x=-\frac{\sqrt{10}}{5}+4$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 3,5 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(x-5\right)\left(x-3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x-5,3.

$$\left(3x-15\right)\left(x-2\right)-\left(3x-9\right)\left(x-4\right)=10\left(x-5\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-15 ki te x-2 ka whakakotahi i ngā kupu rite.

$$3x^{2}-21x+30-\left(3x-9\right)\left(x-4\right)=10\left(x-5\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-9 ki te x-4 ka whakakotahi i ngā kupu rite.

$$3x^{2}-21x+30-\left(3x^{2}-21x+36\right)=10\left(x-5\right)\left(x-3\right)$$

Hei kimi i te tauaro o 3x^{2}-21x+36, kimihia te tauaro o ia taurangi.

$$3x^{2}-21x+30-3x^{2}+21x-36=10\left(x-5\right)\left(x-3\right)$$

Pahekotia te 3x^{2} me -3x^{2}, ka 0.

$$-21x+30+21x-36=10\left(x-5\right)\left(x-3\right)$$

Pahekotia te -21x me 21x, ka 0.

$$30-36=10\left(x-5\right)\left(x-3\right)$$

Tangohia te 36 i te 30, ka -6.

$$-6=10\left(x-5\right)\left(x-3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10 ki te x-5.

$$-6=\left(10x-50\right)\left(x-3\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 10x-50 ki te x-3 ka whakakotahi i ngā kupu rite.

$$-6=10x^{2}-80x+150$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$10x^{2}-80x+150=-6$$

Tangohia te 150 mai i ngā taha e rua.

$$10x^{2}-80x=-6-150$$

Tangohia te 150 i te -6, ka -156.

$$10x^{2}-80x=-156$$

Whakawehea ngā taha e rua ki te 10.

$$\frac{10x^{2}-80x}{10}=-\frac{156}{10}$$

Mā te whakawehe ki te 10 ka wetekia te whakareanga ki te 10.

$$x^{2}+\left(-\frac{80}{10}\right)x=-\frac{156}{10}$$

Whakawehe -80 ki te 10.

$$x^{2}-8x=-\frac{156}{10}$$

Whakahekea te hautanga \frac{-156}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-8x=-\frac{78}{5}$$

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

$$x^{2}-8x+\left(-4\right)^{2}=-\frac{78}{5}+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=-\frac{78}{5}+16$$

Tāpiri -\frac{78}{5} ki te 16.

$$x^{2}-8x+16=\frac{2}{5}$$

Tauwehea x^{2}-8x+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}.

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

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

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

Whakarūnātia.

$$x-4=\frac{\sqrt{10}}{5} x-4=-\frac{\sqrt{10}}{5}$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{10}}{5}+4 x=-\frac{\sqrt{10}}{5}+4$$