Whakaoti i te 35x(765-85x)=405

$$35x(765-85x)=405$$

Whakaoti mō x

$$x=\frac{9\sqrt{351645}}{1190}+\frac{9}{2}\approx 8.984848442$$

$$x=-\frac{9\sqrt{351645}}{1190}+\frac{9}{2}\approx 0.015151558$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 35x ki te 765-85x.

$$26775x-2975x^{2}=405$$

Tangohia te 405 mai i ngā taha e rua.

$$26775x-2975x^{2}-405=0$$

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.

$$-2975x^{2}+26775x-405=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2975 mō a, 26775 mō b, me -405 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-26775±\sqrt{26775^{2}-4\left(-2975\right)\left(-405\right)}}{2\left(-2975\right)}$$

Pūrua 26775.

$$x=\frac{-26775±\sqrt{716900625-4\left(-2975\right)\left(-405\right)}}{2\left(-2975\right)}$$

Whakareatia -4 ki te -2975.

$$x=\frac{-26775±\sqrt{716900625+11900\left(-405\right)}}{2\left(-2975\right)}$$

Whakareatia 11900 ki te -405.

$$x=\frac{-26775±\sqrt{716900625-4819500}}{2\left(-2975\right)}$$

Tāpiri 716900625 ki te -4819500.

$$x=\frac{-26775±\sqrt{712081125}}{2\left(-2975\right)}$$

Tuhia te pūtakerua o te 712081125.

$$x=\frac{-26775±45\sqrt{351645}}{2\left(-2975\right)}$$

Whakareatia 2 ki te -2975.

$$x=\frac{-26775±45\sqrt{351645}}{-5950}$$

Nā, me whakaoti te whārite x=\frac{-26775±45\sqrt{351645}}{-5950} ina he tāpiri te ±. Tāpiri -26775 ki te 45\sqrt{351645}.

$$x=\frac{45\sqrt{351645}-26775}{-5950}$$

Whakawehe -26775+45\sqrt{351645} ki te -5950.

$$x=-\frac{9\sqrt{351645}}{1190}+\frac{9}{2}$$

Nā, me whakaoti te whārite x=\frac{-26775±45\sqrt{351645}}{-5950} ina he tango te ±. Tango 45\sqrt{351645} mai i -26775.

$$x=\frac{-45\sqrt{351645}-26775}{-5950}$$

Whakawehe -26775-45\sqrt{351645} ki te -5950.

$$x=\frac{9\sqrt{351645}}{1190}+\frac{9}{2}$$

Kua oti te whārite te whakatau.

$$x=-\frac{9\sqrt{351645}}{1190}+\frac{9}{2} x=\frac{9\sqrt{351645}}{1190}+\frac{9}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 35x ki te 765-85x.

$$26775x-2975x^{2}=405$$

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.

$$-2975x^{2}+26775x=405$$

Whakawehea ngā taha e rua ki te -2975.

$$\frac{-2975x^{2}+26775x}{-2975}=\frac{405}{-2975}$$

Mā te whakawehe ki te -2975 ka wetekia te whakareanga ki te -2975.

$$x^{2}+\frac{26775}{-2975}x=\frac{405}{-2975}$$

Whakawehe 26775 ki te -2975.

$$x^{2}-9x=\frac{405}{-2975}$$

Whakahekea te hautanga \frac{405}{-2975} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}-9x=-\frac{81}{595}$$

Whakawehea te -9, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{2}. Nā, tāpiria te pūrua o te -\frac{9}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-\frac{81}{595}+\left(-\frac{9}{2}\right)^{2}$$

Pūruatia -\frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-9x+\frac{81}{4}=-\frac{81}{595}+\frac{81}{4}$$

Tāpiri -\frac{81}{595} ki te \frac{81}{4} 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.

$$x^{2}-9x+\frac{81}{4}=\frac{47871}{2380}$$

Tauwehea x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{47871}{2380}$$

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

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{47871}{2380}}$$

Whakarūnātia.

$$x-\frac{9}{2}=\frac{9\sqrt{351645}}{1190} x-\frac{9}{2}=-\frac{9\sqrt{351645}}{1190}$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$x=\frac{9\sqrt{351645}}{1190}+\frac{9}{2} x=-\frac{9\sqrt{351645}}{1190}+\frac{9}{2}$$