Whakaoti i te 9x/x+1-9/x^2+x=50

$$\frac { 9 x } { x + 1 } - \frac { 9 } { x ^ { 2 } + x } = 50$$

Whakaoti mō x

$$x=-\frac{9}{41}\approx -0.219512195$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+1,x^{2}+x.

$$x\times 9x-9=50x\left(x+1\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}\times 9-9=50x\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 50x ki te x+1.

$$x^{2}\times 9-9=50x^{2}+50x$$

Tangohia te 50x^{2} mai i ngā taha e rua.

$$x^{2}\times 9-9-50x^{2}=50x$$

Pahekotia te x^{2}\times 9 me -50x^{2}, ka -41x^{2}.

$$-41x^{2}-9=50x$$

Tangohia te 50x mai i ngā taha e rua.

$$-41x^{2}-9-50x=0$$

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

$$-41x^{2}-50x-9=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -41x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-50 ab=-41\left(-9\right)=369$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 369.

$$-1,-369 -3,-123 -9,-41$$

Tātaihia te tapeke mō ia takirua.

$$-1-369=-370 -3-123=-126 -9-41=-50$$

Ko te otinga te takirua ka hoatu i te tapeke -50.

$$a=-9 b=-41$$

Tuhia anō te -41x^{2}-50x-9 hei \left(-41x^{2}-9x\right)+\left(-41x-9\right).

$$\left(-41x^{2}-9x\right)+\left(-41x-9\right)$$

Tauwehea te -x i te tuatahi me te -1 i te rōpū tuarua.

$$-x\left(41x+9\right)-\left(41x+9\right)$$

Whakatauwehea atu te kīanga pātahi 41x+9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(41x+9\right)\left(-x-1\right)$$

Hei kimi otinga whārite, me whakaoti te 41x+9=0 me te -x-1=0.

$$x=-\frac{9}{41} x=-1$$

Tē taea kia ōrite te tāupe x ki -1.

$$x=-\frac{9}{41}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+1,x^{2}+x.

$$x\times 9x-9=50x\left(x+1\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}\times 9-9=50x\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 50x ki te x+1.

$$x^{2}\times 9-9=50x^{2}+50x$$

Tangohia te 50x^{2} mai i ngā taha e rua.

$$x^{2}\times 9-9-50x^{2}=50x$$

Pahekotia te x^{2}\times 9 me -50x^{2}, ka -41x^{2}.

$$-41x^{2}-9=50x$$

Tangohia te 50x mai i ngā taha e rua.

$$-41x^{2}-9-50x=0$$

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.

$$-41x^{2}-50x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -41 mō a, -50 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\left(-41\right)\left(-9\right)}}{2\left(-41\right)}$$

Pūrua -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\left(-41\right)\left(-9\right)}}{2\left(-41\right)}$$

Whakareatia -4 ki te -41.

$$x=\frac{-\left(-50\right)±\sqrt{2500+164\left(-9\right)}}{2\left(-41\right)}$$

Whakareatia 164 ki te -9.

$$x=\frac{-\left(-50\right)±\sqrt{2500-1476}}{2\left(-41\right)}$$

Tāpiri 2500 ki te -1476.

$$x=\frac{-\left(-50\right)±\sqrt{1024}}{2\left(-41\right)}$$

Tuhia te pūtakerua o te 1024.

$$x=\frac{-\left(-50\right)±32}{2\left(-41\right)}$$

Ko te tauaro o -50 ko 50.

$$x=\frac{50±32}{2\left(-41\right)}$$

Whakareatia 2 ki te -41.

$$x=\frac{50±32}{-82}$$

Nā, me whakaoti te whārite x=\frac{50±32}{-82} ina he tāpiri te ±. Tāpiri 50 ki te 32.

$$x=\frac{82}{-82}$$

Whakawehe 82 ki te -82.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{50±32}{-82} ina he tango te ±. Tango 32 mai i 50.

$$x=\frac{18}{-82}$$

Whakahekea te hautanga \frac{18}{-82} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{9}{41}$$

Kua oti te whārite te whakatau.

$$x=-1 x=-\frac{9}{41}$$

Tē taea kia ōrite te tāupe x ki -1.

$$x=-\frac{9}{41}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+1,x^{2}+x.

$$x\times 9x-9=50x\left(x+1\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{2}\times 9-9=50x\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 50x ki te x+1.

$$x^{2}\times 9-9=50x^{2}+50x$$

Tangohia te 50x^{2} mai i ngā taha e rua.

$$x^{2}\times 9-9-50x^{2}=50x$$

Pahekotia te x^{2}\times 9 me -50x^{2}, ka -41x^{2}.

$$-41x^{2}-9=50x$$

Tangohia te 50x mai i ngā taha e rua.

$$-41x^{2}-9-50x=0$$

Me tāpiri te 9 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-41x^{2}-50x=9$$

Whakawehea ngā taha e rua ki te -41.

$$\frac{-41x^{2}-50x}{-41}=\frac{9}{-41}$$

Mā te whakawehe ki te -41 ka wetekia te whakareanga ki te -41.

$$x^{2}+\left(-\frac{50}{-41}\right)x=\frac{9}{-41}$$

Whakawehe -50 ki te -41.

$$x^{2}+\frac{50}{41}x=\frac{9}{-41}$$

Whakawehe 9 ki te -41.

$$x^{2}+\frac{50}{41}x=-\frac{9}{41}$$

Whakawehea te \frac{50}{41}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{25}{41}. Nā, tāpiria te pūrua o te \frac{25}{41} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{50}{41}x+\left(\frac{25}{41}\right)^{2}=-\frac{9}{41}+\left(\frac{25}{41}\right)^{2}$$

Pūruatia \frac{25}{41} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{50}{41}x+\frac{625}{1681}=-\frac{9}{41}+\frac{625}{1681}$$

Tāpiri -\frac{9}{41} ki te \frac{625}{1681} 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.

$$x^{2}+\frac{50}{41}x+\frac{625}{1681}=\frac{256}{1681}$$

Tauwehea x^{2}+\frac{50}{41}x+\frac{625}{1681}. 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}.

$$\left(x+\frac{25}{41}\right)^{2}=\frac{256}{1681}$$

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

$$\sqrt{\left(x+\frac{25}{41}\right)^{2}}=\sqrt{\frac{256}{1681}}$$

Whakarūnātia.

$$x+\frac{25}{41}=\frac{16}{41} x+\frac{25}{41}=-\frac{16}{41}$$

Me tango \frac{25}{41} mai i ngā taha e rua o te whārite.

$$x=-\frac{9}{41} x=-1$$

Tē taea kia ōrite te tāupe x ki -1.

$$x=-\frac{9}{41}$$