Whakaoti i te 31x^2-3x+1=0

$$31 x ^ { 2 } - 3 x + 1 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{3+\sqrt{115}i}{62}\approx 0.048387097+0.172964602i$$

$$x=\frac{-\sqrt{115}i+3}{62}\approx 0.048387097-0.172964602i$$

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.

$$31x^{2}-3x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 31 mō a, -3 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 31}}{2\times 31}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 31}}{2\times 31}$$

Whakareatia -4 ki te 31.

$$x=\frac{-\left(-3\right)±\sqrt{9-124}}{2\times 31}$$

Tāpiri 9 ki te -124.

$$x=\frac{-\left(-3\right)±\sqrt{-115}}{2\times 31}$$

Tuhia te pūtakerua o te -115.

$$x=\frac{-\left(-3\right)±\sqrt{115}i}{2\times 31}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±\sqrt{115}i}{2\times 31}$$

Whakareatia 2 ki te 31.

$$x=\frac{3±\sqrt{115}i}{62}$$

Nā, me whakaoti te whārite x=\frac{3±\sqrt{115}i}{62} ina he tāpiri te ±. Tāpiri 3 ki te i\sqrt{115}.

$$x=\frac{3+\sqrt{115}i}{62}$$

Nā, me whakaoti te whārite x=\frac{3±\sqrt{115}i}{62} ina he tango te ±. Tango i\sqrt{115} mai i 3.

$$x=\frac{-\sqrt{115}i+3}{62}$$

Kua oti te whārite te whakatau.

$$x=\frac{3+\sqrt{115}i}{62} x=\frac{-\sqrt{115}i+3}{62}$$

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.

$$31x^{2}-3x+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$31x^{2}-3x+1-1=-1$$

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$31x^{2}-3x=-1$$

Whakawehea ngā taha e rua ki te 31.

$$\frac{31x^{2}-3x}{31}=-\frac{1}{31}$$

Mā te whakawehe ki te 31 ka wetekia te whakareanga ki te 31.

$$x^{2}-\frac{3}{31}x=-\frac{1}{31}$$

Whakawehea te -\frac{3}{31}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{62}. Nā, tāpiria te pūrua o te -\frac{3}{62} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{31}x+\left(-\frac{3}{62}\right)^{2}=-\frac{1}{31}+\left(-\frac{3}{62}\right)^{2}$$

Pūruatia -\frac{3}{62} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{31}x+\frac{9}{3844}=-\frac{1}{31}+\frac{9}{3844}$$

Tāpiri -\frac{1}{31} ki te \frac{9}{3844} 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.

$$x^{2}-\frac{3}{31}x+\frac{9}{3844}=-\frac{115}{3844}$$

Tauwehea x^{2}-\frac{3}{31}x+\frac{9}{3844}. 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}.

$$\left(x-\frac{3}{62}\right)^{2}=-\frac{115}{3844}$$

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

$$\sqrt{\left(x-\frac{3}{62}\right)^{2}}=\sqrt{-\frac{115}{3844}}$$

Whakarūnātia.

$$x-\frac{3}{62}=\frac{\sqrt{115}i}{62} x-\frac{3}{62}=-\frac{\sqrt{115}i}{62}$$

Me tāpiri \frac{3}{62} ki ngā taha e rua o te whārite.

$$x=\frac{3+\sqrt{115}i}{62} x=\frac{-\sqrt{115}i+3}{62}$$