Whakaoti i te 50=0.35*40x(1-0.8540x/5)

$$50 = 0.35 \times 40 x ( 1 - 0.85 \frac { 40 x } { 5 } )$$

Whakaoti mō x (complex solution)

$$x=\frac{i\times 5\sqrt{4711}}{476}+\frac{5}{68}\approx 0.073529412+0.720974001i$$

$$x=-\frac{i\times 5\sqrt{4711}}{476}+\frac{5}{68}\approx 0.073529412-0.720974001i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia ngā taha e rua o te whārite ki te 5.

$$250=1.75\times 40x\left(1-0.85\times \frac{40x}{5}\right)$$

Whakareatia te 1.75 ki te 40, ka 70.

$$250=70x\left(1-0.85\times \frac{40x}{5}\right)$$

Whakawehea te 40x ki te 5, kia riro ko 8x.

$$250=70x\left(1-0.85\times 8x\right)$$

Whakareatia te 0.85 ki te 8, ka 6.8.

$$250=70x\left(1-6.8x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 70x ki te 1-6.8x.

$$250=70x-476x^{2}$$

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

$$70x-476x^{2}=250$$

Tangohia te 250 mai i ngā taha e rua.

$$70x-476x^{2}-250=0$$

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.

$$-476x^{2}+70x-250=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -476 mō a, 70 mō b, me -250 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-70±\sqrt{70^{2}-4\left(-476\right)\left(-250\right)}}{2\left(-476\right)}$$

Pūrua 70.

$$x=\frac{-70±\sqrt{4900-4\left(-476\right)\left(-250\right)}}{2\left(-476\right)}$$

Whakareatia -4 ki te -476.

$$x=\frac{-70±\sqrt{4900+1904\left(-250\right)}}{2\left(-476\right)}$$

Whakareatia 1904 ki te -250.

$$x=\frac{-70±\sqrt{4900-476000}}{2\left(-476\right)}$$

Tāpiri 4900 ki te -476000.

$$x=\frac{-70±\sqrt{-471100}}{2\left(-476\right)}$$

Tuhia te pūtakerua o te -471100.

$$x=\frac{-70±10\sqrt{4711}i}{2\left(-476\right)}$$

Whakareatia 2 ki te -476.

$$x=\frac{-70±10\sqrt{4711}i}{-952}$$

Nā, me whakaoti te whārite x=\frac{-70±10\sqrt{4711}i}{-952} ina he tāpiri te ±. Tāpiri -70 ki te 10i\sqrt{4711}.

$$x=\frac{-70+10\sqrt{4711}i}{-952}$$

Whakawehe -70+10i\sqrt{4711} ki te -952.

$$x=-\frac{5\sqrt{4711}i}{476}+\frac{5}{68}$$

Nā, me whakaoti te whārite x=\frac{-70±10\sqrt{4711}i}{-952} ina he tango te ±. Tango 10i\sqrt{4711} mai i -70.

$$x=\frac{-10\sqrt{4711}i-70}{-952}$$

Whakawehe -70-10i\sqrt{4711} ki te -952.

$$x=\frac{5\sqrt{4711}i}{476}+\frac{5}{68}$$

Kua oti te whārite te whakatau.

$$x=-\frac{5\sqrt{4711}i}{476}+\frac{5}{68} x=\frac{5\sqrt{4711}i}{476}+\frac{5}{68}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia ngā taha e rua o te whārite ki te 5.

$$250=1.75\times 40x\left(1-0.85\times \frac{40x}{5}\right)$$

Whakareatia te 1.75 ki te 40, ka 70.

$$250=70x\left(1-0.85\times \frac{40x}{5}\right)$$

Whakawehea te 40x ki te 5, kia riro ko 8x.

$$250=70x\left(1-0.85\times 8x\right)$$

Whakareatia te 0.85 ki te 8, ka 6.8.

$$250=70x\left(1-6.8x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 70x ki te 1-6.8x.

$$250=70x-476x^{2}$$

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

$$70x-476x^{2}=250$$

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.

$$-476x^{2}+70x=250$$

Whakawehea ngā taha e rua ki te -476.

$$\frac{-476x^{2}+70x}{-476}=\frac{250}{-476}$$

Mā te whakawehe ki te -476 ka wetekia te whakareanga ki te -476.

$$x^{2}+\frac{70}{-476}x=\frac{250}{-476}$$

Whakahekea te hautanga \frac{70}{-476} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

$$x^{2}-\frac{5}{34}x=\frac{250}{-476}$$

Whakahekea te hautanga \frac{250}{-476} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{5}{34}x=-\frac{125}{238}$$

Whakawehea te -\frac{5}{34}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{68}. Nā, tāpiria te pūrua o te -\frac{5}{68} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{34}x+\left(-\frac{5}{68}\right)^{2}=-\frac{125}{238}+\left(-\frac{5}{68}\right)^{2}$$

Pūruatia -\frac{5}{68} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{34}x+\frac{25}{4624}=-\frac{125}{238}+\frac{25}{4624}$$

Tāpiri -\frac{125}{238} ki te \frac{25}{4624} 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.

$$x^{2}-\frac{5}{34}x+\frac{25}{4624}=-\frac{16825}{32368}$$

Tauwehea x^{2}-\frac{5}{34}x+\frac{25}{4624}. 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}.

$$\left(x-\frac{5}{68}\right)^{2}=-\frac{16825}{32368}$$

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

$$\sqrt{\left(x-\frac{5}{68}\right)^{2}}=\sqrt{-\frac{16825}{32368}}$$

Whakarūnātia.

$$x-\frac{5}{68}=\frac{5\sqrt{4711}i}{476} x-\frac{5}{68}=-\frac{5\sqrt{4711}i}{476}$$

Me tāpiri \frac{5}{68} ki ngā taha e rua o te whārite.

$$x=\frac{5\sqrt{4711}i}{476}+\frac{5}{68} x=-\frac{5\sqrt{4711}i}{476}+\frac{5}{68}$$