Whakaoti i te 11x^2-10x+13=0

$$11 x ^ { 2 } - 10 x + 13 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{5+\sqrt{118}i}{11}\approx 0.454545455+0.987525499i$$

$$x=\frac{-\sqrt{118}i+5}{11}\approx 0.454545455-0.987525499i$$

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.

$$11x^{2}-10x+13=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 11 mō a, -10 mō b, me 13 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 11\times 13}}{2\times 11}$$

Pūrua -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 11\times 13}}{2\times 11}$$

Whakareatia -4 ki te 11.

$$x=\frac{-\left(-10\right)±\sqrt{100-44\times 13}}{2\times 11}$$

Whakareatia -44 ki te 13.

$$x=\frac{-\left(-10\right)±\sqrt{100-572}}{2\times 11}$$

Tāpiri 100 ki te -572.

$$x=\frac{-\left(-10\right)±\sqrt{-472}}{2\times 11}$$

Tuhia te pūtakerua o te -472.

$$x=\frac{-\left(-10\right)±2\sqrt{118}i}{2\times 11}$$

Ko te tauaro o -10 ko 10.

$$x=\frac{10±2\sqrt{118}i}{2\times 11}$$

Whakareatia 2 ki te 11.

$$x=\frac{10±2\sqrt{118}i}{22}$$

Nā, me whakaoti te whārite x=\frac{10±2\sqrt{118}i}{22} ina he tāpiri te ±. Tāpiri 10 ki te 2i\sqrt{118}.

$$x=\frac{10+2\sqrt{118}i}{22}$$

Whakawehe 10+2i\sqrt{118} ki te 22.

$$x=\frac{5+\sqrt{118}i}{11}$$

Nā, me whakaoti te whārite x=\frac{10±2\sqrt{118}i}{22} ina he tango te ±. Tango 2i\sqrt{118} mai i 10.

$$x=\frac{-2\sqrt{118}i+10}{22}$$

Whakawehe 10-2i\sqrt{118} ki te 22.

$$x=\frac{-\sqrt{118}i+5}{11}$$

Kua oti te whārite te whakatau.

$$x=\frac{5+\sqrt{118}i}{11} x=\frac{-\sqrt{118}i+5}{11}$$

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.

$$11x^{2}-10x+13=0$$

Me tango 13 mai i ngā taha e rua o te whārite.

$$11x^{2}-10x+13-13=-13$$

Mā te tango i te 13 i a ia ake anō ka toe ko te 0.

$$11x^{2}-10x=-13$$

Whakawehea ngā taha e rua ki te 11.

$$\frac{11x^{2}-10x}{11}=-\frac{13}{11}$$

Mā te whakawehe ki te 11 ka wetekia te whakareanga ki te 11.

$$x^{2}-\frac{10}{11}x=-\frac{13}{11}$$

Whakawehea te -\frac{10}{11}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{11}. Nā, tāpiria te pūrua o te -\frac{5}{11} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{10}{11}x+\left(-\frac{5}{11}\right)^{2}=-\frac{13}{11}+\left(-\frac{5}{11}\right)^{2}$$

Pūruatia -\frac{5}{11} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{10}{11}x+\frac{25}{121}=-\frac{13}{11}+\frac{25}{121}$$

Tāpiri -\frac{13}{11} ki te \frac{25}{121} 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.

$$x^{2}-\frac{10}{11}x+\frac{25}{121}=-\frac{118}{121}$$

Tauwehea x^{2}-\frac{10}{11}x+\frac{25}{121}. 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}.

$$\left(x-\frac{5}{11}\right)^{2}=-\frac{118}{121}$$

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

$$\sqrt{\left(x-\frac{5}{11}\right)^{2}}=\sqrt{-\frac{118}{121}}$$

Whakarūnātia.

$$x-\frac{5}{11}=\frac{\sqrt{118}i}{11} x-\frac{5}{11}=-\frac{\sqrt{118}i}{11}$$

Me tāpiri \frac{5}{11} ki ngā taha e rua o te whārite.

$$x=\frac{5+\sqrt{118}i}{11} x=\frac{-\sqrt{118}i+5}{11}$$