Whakaoti i te 219x^2-12x+4=0

$$219 x ^ { 2 } - 12 x + 4 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{2\sqrt{210}i}{219}+\frac{2}{73}\approx 0.02739726+0.13234134i$$

$$x=-\frac{2\sqrt{210}i}{219}+\frac{2}{73}\approx 0.02739726-0.13234134i$$

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.

$$219x^{2}-12x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 219 mō a, -12 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 219\times 4}}{2\times 219}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 219\times 4}}{2\times 219}$$

Whakareatia -4 ki te 219.

$$x=\frac{-\left(-12\right)±\sqrt{144-876\times 4}}{2\times 219}$$

Whakareatia -876 ki te 4.

$$x=\frac{-\left(-12\right)±\sqrt{144-3504}}{2\times 219}$$

Tāpiri 144 ki te -3504.

$$x=\frac{-\left(-12\right)±\sqrt{-3360}}{2\times 219}$$

Tuhia te pūtakerua o te -3360.

$$x=\frac{-\left(-12\right)±4\sqrt{210}i}{2\times 219}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±4\sqrt{210}i}{2\times 219}$$

Whakareatia 2 ki te 219.

$$x=\frac{12±4\sqrt{210}i}{438}$$

Nā, me whakaoti te whārite x=\frac{12±4\sqrt{210}i}{438} ina he tāpiri te ±. Tāpiri 12 ki te 4i\sqrt{210}.

$$x=\frac{12+4\sqrt{210}i}{438}$$

Whakawehe 12+4i\sqrt{210} ki te 438.

$$x=\frac{2\sqrt{210}i}{219}+\frac{2}{73}$$

Nā, me whakaoti te whārite x=\frac{12±4\sqrt{210}i}{438} ina he tango te ±. Tango 4i\sqrt{210} mai i 12.

$$x=\frac{-4\sqrt{210}i+12}{438}$$

Whakawehe 12-4i\sqrt{210} ki te 438.

$$x=-\frac{2\sqrt{210}i}{219}+\frac{2}{73}$$

Kua oti te whārite te whakatau.

$$x=\frac{2\sqrt{210}i}{219}+\frac{2}{73} x=-\frac{2\sqrt{210}i}{219}+\frac{2}{73}$$

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.

$$219x^{2}-12x+4=0$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$219x^{2}-12x+4-4=-4$$

Mā te tango i te 4 i a ia ake anō ka toe ko te 0.

$$219x^{2}-12x=-4$$

Whakawehea ngā taha e rua ki te 219.

$$\frac{219x^{2}-12x}{219}=-\frac{4}{219}$$

Mā te whakawehe ki te 219 ka wetekia te whakareanga ki te 219.

$$x^{2}+\left(-\frac{12}{219}\right)x=-\frac{4}{219}$$

Whakahekea te hautanga \frac{-12}{219} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{4}{73}x=-\frac{4}{219}$$

Whakawehea te -\frac{4}{73}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{2}{73}. Nā, tāpiria te pūrua o te -\frac{2}{73} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{4}{73}x+\left(-\frac{2}{73}\right)^{2}=-\frac{4}{219}+\left(-\frac{2}{73}\right)^{2}$$

Pūruatia -\frac{2}{73} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{4}{73}x+\frac{4}{5329}=-\frac{4}{219}+\frac{4}{5329}$$

Tāpiri -\frac{4}{219} ki te \frac{4}{5329} 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.

$$x^{2}-\frac{4}{73}x+\frac{4}{5329}=-\frac{280}{15987}$$

Tauwehea x^{2}-\frac{4}{73}x+\frac{4}{5329}. 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}.

$$\left(x-\frac{2}{73}\right)^{2}=-\frac{280}{15987}$$

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

$$\sqrt{\left(x-\frac{2}{73}\right)^{2}}=\sqrt{-\frac{280}{15987}}$$

Whakarūnātia.

$$x-\frac{2}{73}=\frac{2\sqrt{210}i}{219} x-\frac{2}{73}=-\frac{2\sqrt{210}i}{219}$$

Me tāpiri \frac{2}{73} ki ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{210}i}{219}+\frac{2}{73} x=-\frac{2\sqrt{210}i}{219}+\frac{2}{73}$$