Whakaoti i te 100x^2-50x+18=0

$$100 x ^ { 2 } - 50 x + 18 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{\sqrt{47}i}{20}+\frac{1}{4}\approx 0.25+0.34278273i$$

$$x=-\frac{\sqrt{47}i}{20}+\frac{1}{4}\approx 0.25-0.34278273i$$

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.

$$100x^{2}-50x+18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 100 mō a, -50 mō b, me 18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 100\times 18}}{2\times 100}$$

Pūrua -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\times 100\times 18}}{2\times 100}$$

Whakareatia -4 ki te 100.

$$x=\frac{-\left(-50\right)±\sqrt{2500-400\times 18}}{2\times 100}$$

Whakareatia -400 ki te 18.

$$x=\frac{-\left(-50\right)±\sqrt{2500-7200}}{2\times 100}$$

Tāpiri 2500 ki te -7200.

$$x=\frac{-\left(-50\right)±\sqrt{-4700}}{2\times 100}$$

Tuhia te pūtakerua o te -4700.

$$x=\frac{-\left(-50\right)±10\sqrt{47}i}{2\times 100}$$

Ko te tauaro o -50 ko 50.

$$x=\frac{50±10\sqrt{47}i}{2\times 100}$$

Whakareatia 2 ki te 100.

$$x=\frac{50±10\sqrt{47}i}{200}$$

Nā, me whakaoti te whārite x=\frac{50±10\sqrt{47}i}{200} ina he tāpiri te ±. Tāpiri 50 ki te 10i\sqrt{47}.

$$x=\frac{50+10\sqrt{47}i}{200}$$

Whakawehe 50+10i\sqrt{47} ki te 200.

$$x=\frac{\sqrt{47}i}{20}+\frac{1}{4}$$

Nā, me whakaoti te whārite x=\frac{50±10\sqrt{47}i}{200} ina he tango te ±. Tango 10i\sqrt{47} mai i 50.

$$x=\frac{-10\sqrt{47}i+50}{200}$$

Whakawehe 50-10i\sqrt{47} ki te 200.

$$x=-\frac{\sqrt{47}i}{20}+\frac{1}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{47}i}{20}+\frac{1}{4} x=-\frac{\sqrt{47}i}{20}+\frac{1}{4}$$

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.

$$100x^{2}-50x+18=0$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$100x^{2}-50x+18-18=-18$$

Mā te tango i te 18 i a ia ake anō ka toe ko te 0.

$$100x^{2}-50x=-18$$

Whakawehea ngā taha e rua ki te 100.

$$\frac{100x^{2}-50x}{100}=-\frac{18}{100}$$

Mā te whakawehe ki te 100 ka wetekia te whakareanga ki te 100.

$$x^{2}+\left(-\frac{50}{100}\right)x=-\frac{18}{100}$$

Whakahekea te hautanga \frac{-50}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 50.

$$x^{2}-\frac{1}{2}x=-\frac{18}{100}$$

Whakahekea te hautanga \frac{-18}{100} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{1}{2}x=-\frac{9}{50}$$

Whakawehea te -\frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{4}. Nā, tāpiria te pūrua o te -\frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=-\frac{9}{50}+\left(-\frac{1}{4}\right)^{2}$$

Pūruatia -\frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{9}{50}+\frac{1}{16}$$

Tāpiri -\frac{9}{50} ki te \frac{1}{16} 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.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{47}{400}$$

Tauwehea x^{2}-\frac{1}{2}x+\frac{1}{16}. 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}.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{47}{400}$$

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

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{47}{400}}$$

Whakarūnātia.

$$x-\frac{1}{4}=\frac{\sqrt{47}i}{20} x-\frac{1}{4}=-\frac{\sqrt{47}i}{20}$$

Me tāpiri \frac{1}{4} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{47}i}{20}+\frac{1}{4} x=-\frac{\sqrt{47}i}{20}+\frac{1}{4}$$