Whakaoti i te x^-1-90x^-2+1=0

$$x ^ { - 1 } - 90 x ^ { - 2 } + 1 = 0$$

Whakaoti mō x

$$x=-10$$

$$x=9$$

Ngā Upane Whakamahi Tauwehe

Whakaraupapatia anō ngā kīanga tau.

$$1+\frac{1}{x}-90x^{-2}=0$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$x+1-90x^{-2}x=0$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te -2 me te 1 kia riro ai te -1.

$$x+1-90x^{-1}=0$$

Whakaraupapatia anō ngā kīanga tau.

$$x+1-90\times \frac{1}{x}=0$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$xx+x-90=0$$

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

$$x^{2}+x-90=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}+x-90 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=-90$$

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 -90.

$$-1,90 -2,45 -3,30 -5,18 -6,15 -9,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+90=89 -2+45=43 -3+30=27 -5+18=13 -6+15=9 -9+10=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=-9 b=10$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x-9\right)\left(x+10\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x+10=0.

$$x=9 x=-10$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakaraupapatia anō ngā kīanga tau.

$$1+\frac{1}{x}-90x^{-2}=0$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$x+1-90x^{-2}x=0$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te -2 me te 1 kia riro ai te -1.

$$x+1-90x^{-1}=0$$

Whakaraupapatia anō ngā kīanga tau.

$$x+1-90\times \frac{1}{x}=0$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$xx+x-90=0$$

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

$$x^{2}+x-90=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-90. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=1\left(-90\right)=-90$$

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 -90.

$$-1,90 -2,45 -3,30 -5,18 -6,15 -9,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+90=89 -2+45=43 -3+30=27 -5+18=13 -6+15=9 -9+10=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=-9 b=10$$

Tuhia anō te x^{2}+x-90 hei \left(x^{2}-9x\right)+\left(10x-90\right).

$$\left(x^{2}-9x\right)+\left(10x-90\right)$$

Tauwehea te x i te tuatahi me te 10 i te rōpū tuarua.

$$x\left(x-9\right)+10\left(x-9\right)$$

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

$$\left(x-9\right)\left(x+10\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x+10=0.

$$x=9 x=-10$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakaraupapatia anō ngā kīanga tau.

$$1+\frac{1}{x}-90x^{-2}=0$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$x+1-90x^{-2}x=0$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te -2 me te 1 kia riro ai te -1.

$$x+1-90x^{-1}=0$$

Whakaraupapatia anō ngā kīanga tau.

$$x+1-90\times \frac{1}{x}=0$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$xx+x-90=0$$

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

$$x^{2}+x-90=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 1 mō b, me -90 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-90\right)}}{2}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\left(-90\right)}}{2}$$

Whakareatia -4 ki te -90.

$$x=\frac{-1±\sqrt{1+360}}{2}$$

Tāpiri 1 ki te 360.

$$x=\frac{-1±\sqrt{361}}{2}$$

Tuhia te pūtakerua o te 361.

$$x=\frac{-1±19}{2}$$

Nā, me whakaoti te whārite x=\frac{-1±19}{2} ina he tāpiri te ±. Tāpiri -1 ki te 19.

$$x=\frac{18}{2}$$

Whakawehe 18 ki te 2.

$$x=9$$

Nā, me whakaoti te whārite x=\frac{-1±19}{2} ina he tango te ±. Tango 19 mai i -1.

$$x=-\frac{20}{2}$$

Whakawehe -20 ki te 2.

$$x=-10$$

Kua oti te whārite te whakatau.

$$x=9 x=-10$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 1 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$x^{-1}-90x^{-2}=-1$$

Whakaraupapatia anō ngā kīanga tau.

$$\frac{1}{x}-90x^{-2}=-1$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$1-90x^{-2}x=-x$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te -2 me te 1 kia riro ai te -1.

$$1-90x^{-1}=-x$$

Me tāpiri te x ki ngā taha e rua.

$$1-90x^{-1}+x=0$$

Tangohia te 1 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-90x^{-1}+x=-1$$

Whakaraupapatia anō ngā kīanga tau.

$$x-90\times \frac{1}{x}=-1$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$xx-90=-x$$

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

$$x^{2}-90=-x$$

Me tāpiri te x ki ngā taha e rua.

$$x^{2}-90+x=0$$

Me tāpiri te 90 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x^{2}+x=90$$

Whakawehea te 1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{2}. Nā, tāpiria te pūrua o te \frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=90+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+x+\frac{1}{4}=90+\frac{1}{4}$$

Tāpiri 90 ki te \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{361}{4}$$

Tauwehea x^{2}+x+\frac{1}{4}. 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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{361}{4}$$

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

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Whakarūnātia.

$$x+\frac{1}{2}=\frac{19}{2} x+\frac{1}{2}=-\frac{19}{2}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$x=9 x=-10$$