Whakaoti i te 10x^2-x+3=0

$$10 x ^ { 2 } - x + 3 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{1+\sqrt{119}i}{20}\approx 0.05+0.545435606i$$

$$x=\frac{-\sqrt{119}i+1}{20}\approx 0.05-0.545435606i$$

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.

$$10x^{2}-x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 10 mō a, -1 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 10\times 3}}{2\times 10}$$

Whakareatia -4 ki te 10.

$$x=\frac{-\left(-1\right)±\sqrt{1-40\times 3}}{2\times 10}$$

Whakareatia -40 ki te 3.

$$x=\frac{-\left(-1\right)±\sqrt{1-120}}{2\times 10}$$

Tāpiri 1 ki te -120.

$$x=\frac{-\left(-1\right)±\sqrt{-119}}{2\times 10}$$

Tuhia te pūtakerua o te -119.

$$x=\frac{-\left(-1\right)±\sqrt{119}i}{2\times 10}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±\sqrt{119}i}{2\times 10}$$

Whakareatia 2 ki te 10.

$$x=\frac{1±\sqrt{119}i}{20}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{119}i}{20} ina he tāpiri te ±. Tāpiri 1 ki te i\sqrt{119}.

$$x=\frac{1+\sqrt{119}i}{20}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{119}i}{20} ina he tango te ±. Tango i\sqrt{119} mai i 1.

$$x=\frac{-\sqrt{119}i+1}{20}$$

Kua oti te whārite te whakatau.

$$x=\frac{1+\sqrt{119}i}{20} x=\frac{-\sqrt{119}i+1}{20}$$

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.

$$10x^{2}-x+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$10x^{2}-x+3-3=-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$10x^{2}-x=-3$$

Whakawehea ngā taha e rua ki te 10.

$$\frac{10x^{2}-x}{10}=-\frac{3}{10}$$

Mā te whakawehe ki te 10 ka wetekia te whakareanga ki te 10.

$$x^{2}-\frac{1}{10}x=-\frac{3}{10}$$

Whakawehea te -\frac{1}{10}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{20}. Nā, tāpiria te pūrua o te -\frac{1}{20} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{10}x+\left(-\frac{1}{20}\right)^{2}=-\frac{3}{10}+\left(-\frac{1}{20}\right)^{2}$$

Pūruatia -\frac{1}{20} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{10}x+\frac{1}{400}=-\frac{3}{10}+\frac{1}{400}$$

Tāpiri -\frac{3}{10} ki te \frac{1}{400} 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.

$$x^{2}-\frac{1}{10}x+\frac{1}{400}=-\frac{119}{400}$$

Tauwehea x^{2}-\frac{1}{10}x+\frac{1}{400}. 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}.

$$\left(x-\frac{1}{20}\right)^{2}=-\frac{119}{400}$$

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

$$\sqrt{\left(x-\frac{1}{20}\right)^{2}}=\sqrt{-\frac{119}{400}}$$

Whakarūnātia.

$$x-\frac{1}{20}=\frac{\sqrt{119}i}{20} x-\frac{1}{20}=-\frac{\sqrt{119}i}{20}$$

Me tāpiri \frac{1}{20} ki ngā taha e rua o te whārite.

$$x=\frac{1+\sqrt{119}i}{20} x=\frac{-\sqrt{119}i+1}{20}$$