Whakaoti i te 13x^2-5x+4=0

$$13 x ^ { 2 } - 5 x + 4 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{5+\sqrt{183}i}{26}\approx 0.192307692+0.520298048i$$

$$x=\frac{-\sqrt{183}i+5}{26}\approx 0.192307692-0.520298048i$$

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.

$$13x^{2}-5x+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 13 mō a, -5 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 13\times 4}}{2\times 13}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 13\times 4}}{2\times 13}$$

Whakareatia -4 ki te 13.

$$x=\frac{-\left(-5\right)±\sqrt{25-52\times 4}}{2\times 13}$$

Whakareatia -52 ki te 4.

$$x=\frac{-\left(-5\right)±\sqrt{25-208}}{2\times 13}$$

Tāpiri 25 ki te -208.

$$x=\frac{-\left(-5\right)±\sqrt{-183}}{2\times 13}$$

Tuhia te pūtakerua o te -183.

$$x=\frac{-\left(-5\right)±\sqrt{183}i}{2\times 13}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±\sqrt{183}i}{2\times 13}$$

Whakareatia 2 ki te 13.

$$x=\frac{5±\sqrt{183}i}{26}$$

Nā, me whakaoti te whārite x=\frac{5±\sqrt{183}i}{26} ina he tāpiri te ±. Tāpiri 5 ki te i\sqrt{183}.

$$x=\frac{5+\sqrt{183}i}{26}$$

Nā, me whakaoti te whārite x=\frac{5±\sqrt{183}i}{26} ina he tango te ±. Tango i\sqrt{183} mai i 5.

$$x=\frac{-\sqrt{183}i+5}{26}$$

Kua oti te whārite te whakatau.

$$x=\frac{5+\sqrt{183}i}{26} x=\frac{-\sqrt{183}i+5}{26}$$

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.

$$13x^{2}-5x+4=0$$

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

$$13x^{2}-5x+4-4=-4$$

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

$$13x^{2}-5x=-4$$

Whakawehea ngā taha e rua ki te 13.

$$\frac{13x^{2}-5x}{13}=-\frac{4}{13}$$

Mā te whakawehe ki te 13 ka wetekia te whakareanga ki te 13.

$$x^{2}-\frac{5}{13}x=-\frac{4}{13}$$

Whakawehea te -\frac{5}{13}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{26}. Nā, tāpiria te pūrua o te -\frac{5}{26} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{13}x+\left(-\frac{5}{26}\right)^{2}=-\frac{4}{13}+\left(-\frac{5}{26}\right)^{2}$$

Pūruatia -\frac{5}{26} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{13}x+\frac{25}{676}=-\frac{4}{13}+\frac{25}{676}$$

Tāpiri -\frac{4}{13} ki te \frac{25}{676} 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.

$$x^{2}-\frac{5}{13}x+\frac{25}{676}=-\frac{183}{676}$$

Tauwehea x^{2}-\frac{5}{13}x+\frac{25}{676}. 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}.

$$\left(x-\frac{5}{26}\right)^{2}=-\frac{183}{676}$$

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

$$\sqrt{\left(x-\frac{5}{26}\right)^{2}}=\sqrt{-\frac{183}{676}}$$

Whakarūnātia.

$$x-\frac{5}{26}=\frac{\sqrt{183}i}{26} x-\frac{5}{26}=-\frac{\sqrt{183}i}{26}$$

Me tāpiri \frac{5}{26} ki ngā taha e rua o te whārite.

$$x=\frac{5+\sqrt{183}i}{26} x=\frac{-\sqrt{183}i+5}{26}$$