Whakaoti i te 125x^2-11x+10=0

$$125 x ^ { 2 } - 11 x + 10 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{11+\sqrt{4879}i}{250}\approx 0.044+0.279399356i$$

$$x=\frac{-\sqrt{4879}i+11}{250}\approx 0.044-0.279399356i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$125x^{2}-11x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 125 mō a, -11 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 125\times 10}}{2\times 125}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 125\times 10}}{2\times 125}$$

Whakareatia -4 ki te 125.

$$x=\frac{-\left(-11\right)±\sqrt{121-500\times 10}}{2\times 125}$$

Whakareatia -500 ki te 10.

$$x=\frac{-\left(-11\right)±\sqrt{121-5000}}{2\times 125}$$

Tāpiri 121 ki te -5000.

$$x=\frac{-\left(-11\right)±\sqrt{-4879}}{2\times 125}$$

Tuhia te pūtakerua o te -4879.

$$x=\frac{-\left(-11\right)±\sqrt{4879}i}{2\times 125}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±\sqrt{4879}i}{2\times 125}$$

Whakareatia 2 ki te 125.

$$x=\frac{11±\sqrt{4879}i}{250}$$

Nā, me whakaoti te whārite x=\frac{11±\sqrt{4879}i}{250} ina he tāpiri te ±. Tāpiri 11 ki te i\sqrt{4879}.

$$x=\frac{11+\sqrt{4879}i}{250}$$

Nā, me whakaoti te whārite x=\frac{11±\sqrt{4879}i}{250} ina he tango te ±. Tango i\sqrt{4879} mai i 11.

$$x=\frac{-\sqrt{4879}i+11}{250}$$

Kua oti te whārite te whakatau.

$$x=\frac{11+\sqrt{4879}i}{250} x=\frac{-\sqrt{4879}i+11}{250}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$125x^{2}-11x+10=0$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$125x^{2}-11x+10-10=-10$$

Mā te tango i te 10 i a ia ake anō ka toe ko te 0.

$$125x^{2}-11x=-10$$

Whakawehea ngā taha e rua ki te 125.

$$\frac{125x^{2}-11x}{125}=-\frac{10}{125}$$

Mā te whakawehe ki te 125 ka wetekia te whakareanga ki te 125.

$$x^{2}-\frac{11}{125}x=-\frac{10}{125}$$

Whakahekea te hautanga \frac{-10}{125} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}-\frac{11}{125}x=-\frac{2}{25}$$

Whakawehea te -\frac{11}{125}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{250}. Nā, tāpiria te pūrua o te -\frac{11}{250} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{11}{125}x+\left(-\frac{11}{250}\right)^{2}=-\frac{2}{25}+\left(-\frac{11}{250}\right)^{2}$$

Pūruatia -\frac{11}{250} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{11}{125}x+\frac{121}{62500}=-\frac{2}{25}+\frac{121}{62500}$$

Tāpiri -\frac{2}{25} ki te \frac{121}{62500} 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.

$$x^{2}-\frac{11}{125}x+\frac{121}{62500}=-\frac{4879}{62500}$$

Tauwehea x^{2}-\frac{11}{125}x+\frac{121}{62500}. 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}.

$$\left(x-\frac{11}{250}\right)^{2}=-\frac{4879}{62500}$$

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

$$\sqrt{\left(x-\frac{11}{250}\right)^{2}}=\sqrt{-\frac{4879}{62500}}$$

Whakarūnātia.

$$x-\frac{11}{250}=\frac{\sqrt{4879}i}{250} x-\frac{11}{250}=-\frac{\sqrt{4879}i}{250}$$

Me tāpiri \frac{11}{250} ki ngā taha e rua o te whārite.

$$x=\frac{11+\sqrt{4879}i}{250} x=\frac{-\sqrt{4879}i+11}{250}$$