Whakaoti i te x^2-37x+365=0

$${ x }^{ 2 } -37x+365=0$$

Whakaoti mō x (complex solution)

$$x=\frac{37+\sqrt{91}i}{2}\approx 18.5+4.769696007i$$

$$x=\frac{-\sqrt{91}i+37}{2}\approx 18.5-4.769696007i$$

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.

$$x^{2}-37x+365=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -37 mō b, me 365 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-37\right)±\sqrt{\left(-37\right)^{2}-4\times 365}}{2}$$

Pūrua -37.

$$x=\frac{-\left(-37\right)±\sqrt{1369-4\times 365}}{2}$$

Whakareatia -4 ki te 365.

$$x=\frac{-\left(-37\right)±\sqrt{1369-1460}}{2}$$

Tāpiri 1369 ki te -1460.

$$x=\frac{-\left(-37\right)±\sqrt{-91}}{2}$$

Tuhia te pūtakerua o te -91.

$$x=\frac{-\left(-37\right)±\sqrt{91}i}{2}$$

Ko te tauaro o -37 ko 37.

$$x=\frac{37±\sqrt{91}i}{2}$$

Nā, me whakaoti te whārite x=\frac{37±\sqrt{91}i}{2} ina he tāpiri te ±. Tāpiri 37 ki te i\sqrt{91}.

$$x=\frac{37+\sqrt{91}i}{2}$$

Nā, me whakaoti te whārite x=\frac{37±\sqrt{91}i}{2} ina he tango te ±. Tango i\sqrt{91} mai i 37.

$$x=\frac{-\sqrt{91}i+37}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{37+\sqrt{91}i}{2} x=\frac{-\sqrt{91}i+37}{2}$$

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.

$$x^{2}-37x+365=0$$

Me tango 365 mai i ngā taha e rua o te whārite.

$$x^{2}-37x+365-365=-365$$

Mā te tango i te 365 i a ia ake anō ka toe ko te 0.

$$x^{2}-37x=-365$$

Whakawehea te -37, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{37}{2}. Nā, tāpiria te pūrua o te -\frac{37}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-37x+\left(-\frac{37}{2}\right)^{2}=-365+\left(-\frac{37}{2}\right)^{2}$$

Pūruatia -\frac{37}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-37x+\frac{1369}{4}=-365+\frac{1369}{4}$$

Tāpiri -365 ki te \frac{1369}{4}.

$$x^{2}-37x+\frac{1369}{4}=-\frac{91}{4}$$

Tauwehea x^{2}-37x+\frac{1369}{4}. 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}.

$$\left(x-\frac{37}{2}\right)^{2}=-\frac{91}{4}$$

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

$$\sqrt{\left(x-\frac{37}{2}\right)^{2}}=\sqrt{-\frac{91}{4}}$$

Whakarūnātia.

$$x-\frac{37}{2}=\frac{\sqrt{91}i}{2} x-\frac{37}{2}=-\frac{\sqrt{91}i}{2}$$

Me tāpiri \frac{37}{2} ki ngā taha e rua o te whārite.

$$x=\frac{37+\sqrt{91}i}{2} x=\frac{-\sqrt{91}i+37}{2}$$