Whakaoti i te 1/9x^2-6x+1+16/15x^2+25x-10=1/3x-1

$$\frac { 1 } { 9 x ^ { 2 } - 6 x + 1 } + \frac { 16 } { 15 x ^ { 2 } + 25 x - 10 } = \frac { 1 } { 3 x - 1 }$$

Whakaoti mō x

$$x=-\frac{2}{15}\approx -0.133333333$$

$$x=2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 5\left(x+2\right)\left(3x-1\right)^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o 9x^{2}-6x+1,15x^{2}+25x-10,3x-1.

$$5x+10+\left(3x-1\right)\times 16=5\left(x+2\right)\left(3x-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x-1 ki te 16.

$$5x+10+48x-16=5\left(x+2\right)\left(3x-1\right)$$

Pahekotia te 5x me 48x, ka 53x.

$$53x+10-16=5\left(x+2\right)\left(3x-1\right)$$

Tangohia te 16 i te 10, ka -6.

$$53x-6=5\left(x+2\right)\left(3x-1\right)$$

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

$$53x-6=\left(5x+10\right)\left(3x-1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 5x+10 ki te 3x-1 ka whakakotahi i ngā kupu rite.

$$53x-6=15x^{2}+25x-10$$

Tangohia te 15x^{2} mai i ngā taha e rua.

$$53x-6-15x^{2}=25x-10$$

Tangohia te 25x mai i ngā taha e rua.

$$53x-6-15x^{2}-25x=-10$$

Pahekotia te 53x me -25x, ka 28x.

$$28x-6-15x^{2}=-10$$

Me tāpiri te 10 ki ngā taha e rua.

$$28x-6-15x^{2}+10=0$$

Tāpirihia te -6 ki te 10, ka 4.

$$28x+4-15x^{2}=0$$

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

$$-15x^{2}+28x+4=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -15x^{2}+ax+bx+4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=28 ab=-15\times 4=-60$$

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 -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

Ko te otinga te takirua ka hoatu i te tapeke 28.

$$a=30 b=-2$$

Tuhia anō te -15x^{2}+28x+4 hei \left(-15x^{2}+30x\right)+\left(-2x+4\right).

$$\left(-15x^{2}+30x\right)+\left(-2x+4\right)$$

Tauwehea te 15x i te tuatahi me te 2 i te rōpū tuarua.

$$15x\left(-x+2\right)+2\left(-x+2\right)$$

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

$$\left(-x+2\right)\left(15x+2\right)$$

Hei kimi otinga whārite, me whakaoti te -x+2=0 me te 15x+2=0.

$$x=2 x=-\frac{2}{15}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 5\left(x+2\right)\left(3x-1\right)^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o 9x^{2}-6x+1,15x^{2}+25x-10,3x-1.

$$5x+10+\left(3x-1\right)\times 16=5\left(x+2\right)\left(3x-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x-1 ki te 16.

$$5x+10+48x-16=5\left(x+2\right)\left(3x-1\right)$$

Pahekotia te 5x me 48x, ka 53x.

$$53x+10-16=5\left(x+2\right)\left(3x-1\right)$$

Tangohia te 16 i te 10, ka -6.

$$53x-6=5\left(x+2\right)\left(3x-1\right)$$

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

$$53x-6=\left(5x+10\right)\left(3x-1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 5x+10 ki te 3x-1 ka whakakotahi i ngā kupu rite.

$$53x-6=15x^{2}+25x-10$$

Tangohia te 15x^{2} mai i ngā taha e rua.

$$53x-6-15x^{2}=25x-10$$

Tangohia te 25x mai i ngā taha e rua.

$$53x-6-15x^{2}-25x=-10$$

Pahekotia te 53x me -25x, ka 28x.

$$28x-6-15x^{2}=-10$$

Me tāpiri te 10 ki ngā taha e rua.

$$28x-6-15x^{2}+10=0$$

Tāpirihia te -6 ki te 10, ka 4.

$$28x+4-15x^{2}=0$$

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.

$$-15x^{2}+28x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -15 mō a, 28 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-15\right)\times 4}}{2\left(-15\right)}$$

Pūrua 28.

$$x=\frac{-28±\sqrt{784-4\left(-15\right)\times 4}}{2\left(-15\right)}$$

Whakareatia -4 ki te -15.

$$x=\frac{-28±\sqrt{784+60\times 4}}{2\left(-15\right)}$$

Whakareatia 60 ki te 4.

$$x=\frac{-28±\sqrt{784+240}}{2\left(-15\right)}$$

Tāpiri 784 ki te 240.

$$x=\frac{-28±\sqrt{1024}}{2\left(-15\right)}$$

Tuhia te pūtakerua o te 1024.

$$x=\frac{-28±32}{2\left(-15\right)}$$

Whakareatia 2 ki te -15.

$$x=\frac{-28±32}{-30}$$

Nā, me whakaoti te whārite x=\frac{-28±32}{-30} ina he tāpiri te ±. Tāpiri -28 ki te 32.

$$x=\frac{4}{-30}$$

Whakahekea te hautanga \frac{4}{-30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{2}{15}$$

Nā, me whakaoti te whārite x=\frac{-28±32}{-30} ina he tango te ±. Tango 32 mai i -28.

$$x=-\frac{60}{-30}$$

Whakawehe -60 ki te -30.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=-\frac{2}{15} x=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,\frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 5\left(x+2\right)\left(3x-1\right)^{2}, arā, te tauraro pātahi he tino iti rawa te kitea o 9x^{2}-6x+1,15x^{2}+25x-10,3x-1.

$$5x+10+\left(3x-1\right)\times 16=5\left(x+2\right)\left(3x-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x-1 ki te 16.

$$5x+10+48x-16=5\left(x+2\right)\left(3x-1\right)$$

Pahekotia te 5x me 48x, ka 53x.

$$53x+10-16=5\left(x+2\right)\left(3x-1\right)$$

Tangohia te 16 i te 10, ka -6.

$$53x-6=5\left(x+2\right)\left(3x-1\right)$$

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

$$53x-6=\left(5x+10\right)\left(3x-1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 5x+10 ki te 3x-1 ka whakakotahi i ngā kupu rite.

$$53x-6=15x^{2}+25x-10$$

Tangohia te 15x^{2} mai i ngā taha e rua.

$$53x-6-15x^{2}=25x-10$$

Tangohia te 25x mai i ngā taha e rua.

$$53x-6-15x^{2}-25x=-10$$

Pahekotia te 53x me -25x, ka 28x.

$$28x-6-15x^{2}=-10$$

Me tāpiri te 6 ki ngā taha e rua.

$$28x-15x^{2}=-10+6$$

Tāpirihia te -10 ki te 6, ka -4.

$$28x-15x^{2}=-4$$

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.

$$-15x^{2}+28x=-4$$

Whakawehea ngā taha e rua ki te -15.

$$\frac{-15x^{2}+28x}{-15}=-\frac{4}{-15}$$

Mā te whakawehe ki te -15 ka wetekia te whakareanga ki te -15.

$$x^{2}+\frac{28}{-15}x=-\frac{4}{-15}$$

Whakawehe 28 ki te -15.

$$x^{2}-\frac{28}{15}x=-\frac{4}{-15}$$

Whakawehe -4 ki te -15.

$$x^{2}-\frac{28}{15}x=\frac{4}{15}$$

Whakawehea te -\frac{28}{15}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{14}{15}. Nā, tāpiria te pūrua o te -\frac{14}{15} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{28}{15}x+\left(-\frac{14}{15}\right)^{2}=\frac{4}{15}+\left(-\frac{14}{15}\right)^{2}$$

Pūruatia -\frac{14}{15} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{28}{15}x+\frac{196}{225}=\frac{4}{15}+\frac{196}{225}$$

Tāpiri \frac{4}{15} ki te \frac{196}{225} 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.

$$x^{2}-\frac{28}{15}x+\frac{196}{225}=\frac{256}{225}$$

Tauwehea x^{2}-\frac{28}{15}x+\frac{196}{225}. 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}.

$$\left(x-\frac{14}{15}\right)^{2}=\frac{256}{225}$$

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

$$\sqrt{\left(x-\frac{14}{15}\right)^{2}}=\sqrt{\frac{256}{225}}$$

Whakarūnātia.

$$x-\frac{14}{15}=\frac{16}{15} x-\frac{14}{15}=-\frac{16}{15}$$

Me tāpiri \frac{14}{15} ki ngā taha e rua o te whārite.

$$x=2 x=-\frac{2}{15}$$