Whakaoti i te x+1/x=100

$$x + \frac { 1 } { x } = 100$$

Whakaoti mō x

$$x=7\sqrt{51}+50\approx 99.989999$$

$$x=50-7\sqrt{51}\approx 0.010001$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$xx+1=100x$$

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

$$x^{2}+1=100x$$

Tangohia te 100x mai i ngā taha e rua.

$$x^{2}+1-100x=0$$

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.

$$x^{2}-100x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -100 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-100\right)±\sqrt{\left(-100\right)^{2}-4}}{2}$$

Pūrua -100.

$$x=\frac{-\left(-100\right)±\sqrt{10000-4}}{2}$$

Tāpiri 10000 ki te -4.

$$x=\frac{-\left(-100\right)±\sqrt{9996}}{2}$$

Tuhia te pūtakerua o te 9996.

$$x=\frac{-\left(-100\right)±14\sqrt{51}}{2}$$

Ko te tauaro o -100 ko 100.

$$x=\frac{100±14\sqrt{51}}{2}$$

Nā, me whakaoti te whārite x=\frac{100±14\sqrt{51}}{2} ina he tāpiri te ±. Tāpiri 100 ki te 14\sqrt{51}.

$$x=\frac{14\sqrt{51}+100}{2}$$

Whakawehe 100+14\sqrt{51} ki te 2.

$$x=7\sqrt{51}+50$$

Nā, me whakaoti te whārite x=\frac{100±14\sqrt{51}}{2} ina he tango te ±. Tango 14\sqrt{51} mai i 100.

$$x=\frac{100-14\sqrt{51}}{2}$$

Whakawehe 100-14\sqrt{51} ki te 2.

$$x=50-7\sqrt{51}$$

Kua oti te whārite te whakatau.

$$x=7\sqrt{51}+50 x=50-7\sqrt{51}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$xx+1=100x$$

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

$$x^{2}+1=100x$$

Tangohia te 100x mai i ngā taha e rua.

$$x^{2}+1-100x=0$$

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

$$x^{2}-100x=-1$$

Whakawehea te -100, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -50. Nā, tāpiria te pūrua o te -50 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-100x+\left(-50\right)^{2}=-1+\left(-50\right)^{2}$$

Pūrua -50.

$$x^{2}-100x+2500=-1+2500$$

Tāpiri -1 ki te 2500.

$$x^{2}-100x+2500=2499$$

Tauwehea x^{2}-100x+2500. 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}.

$$\left(x-50\right)^{2}=2499$$

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

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{2499}$$

Whakarūnātia.

$$x-50=7\sqrt{51} x-50=-7\sqrt{51}$$

Me tāpiri 50 ki ngā taha e rua o te whārite.

$$x=7\sqrt{51}+50 x=50-7\sqrt{51}$$