Whakaoti i te 1/x=-x+25

$$\frac{ 1 }{ x } = -x+25$$

Whakaoti mō x

$$x = \frac{3 \sqrt{69} + 25}{2} \approx 24.959935794$$

$$x=\frac{25-3\sqrt{69}}{2}\approx 0.040064206$$

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.

$$1=-xx+x\times 25$$

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

$$1=-x^{2}+x\times 25$$

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

$$-x^{2}+x\times 25=1$$

Tangohia te 1 mai i ngā taha e rua.

$$-x^{2}+x\times 25-1=0$$

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.

$$-x^{2}+25x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 25 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25±\sqrt{25^{2}-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Pūrua 25.

$$x=\frac{-25±\sqrt{625-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-25±\sqrt{625+4\left(-1\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -1.

$$x=\frac{-25±\sqrt{625-4}}{2\left(-1\right)}$$

Tāpiri 625 ki te -4.

$$x=\frac{-25±\sqrt{621}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 621.

$$x=\frac{-25±3\sqrt{69}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-25±3\sqrt{69}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-25±3\sqrt{69}}{-2} ina he tāpiri te ±. Tāpiri -25 ki te 3\sqrt{69}.

$$x=\frac{3\sqrt{69}-25}{-2}$$

Whakawehe -25+3\sqrt{69} ki te -2.

$$x=\frac{25-3\sqrt{69}}{2}$$

Nā, me whakaoti te whārite x=\frac{-25±3\sqrt{69}}{-2} ina he tango te ±. Tango 3\sqrt{69} mai i -25.

$$x=\frac{-3\sqrt{69}-25}{-2}$$

Whakawehe -25-3\sqrt{69} ki te -2.

$$x=\frac{3\sqrt{69}+25}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{25-3\sqrt{69}}{2} x=\frac{3\sqrt{69}+25}{2}$$

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.

$$1=-xx+x\times 25$$

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

$$1=-x^{2}+x\times 25$$

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

$$-x^{2}+x\times 25=1$$

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.

$$-x^{2}+25x=1$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+25x}{-1}=\frac{1}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{25}{-1}x=\frac{1}{-1}$$

Whakawehe 25 ki te -1.

$$x^{2}-25x=\frac{1}{-1}$$

Whakawehe 1 ki te -1.

$$x^{2}-25x=-1$$

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

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-1+\left(-\frac{25}{2}\right)^{2}$$

Pūruatia -\frac{25}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-25x+\frac{625}{4}=-1+\frac{625}{4}$$

Tāpiri -1 ki te \frac{625}{4}.

$$x^{2}-25x+\frac{625}{4}=\frac{621}{4}$$

Tauwehea te x^{2}-25x+\frac{625}{4}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{621}{4}$$

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

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{621}{4}}$$

Whakarūnātia.

$$x-\frac{25}{2}=\frac{3\sqrt{69}}{2} x-\frac{25}{2}=-\frac{3\sqrt{69}}{2}$$

Me tāpiri \frac{25}{2} ki ngā taha e rua o te whārite.

$$x=\frac{3\sqrt{69}+25}{2} x=\frac{25-3\sqrt{69}}{2}$$