Whakaoti i te 13158x^2-2756x+27360=0

$$13158 { x }^{ 2 } -2756x+27360=0$$

Whakaoti mō x (complex solution)

$$x=\frac{689+\sqrt{89525999}i}{6579}\approx 0.104727162+1.438184824i$$

$$x=\frac{-\sqrt{89525999}i+689}{6579}\approx 0.104727162-1.438184824i$$

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.

$$13158x^{2}-2756x+27360=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 13158 mō a, -2756 mō b, me 27360 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2756\right)±\sqrt{\left(-2756\right)^{2}-4\times 13158\times 27360}}{2\times 13158}$$

Pūrua -2756.

$$x=\frac{-\left(-2756\right)±\sqrt{7595536-4\times 13158\times 27360}}{2\times 13158}$$

Whakareatia -4 ki te 13158.

$$x=\frac{-\left(-2756\right)±\sqrt{7595536-52632\times 27360}}{2\times 13158}$$

Whakareatia -52632 ki te 27360.

$$x=\frac{-\left(-2756\right)±\sqrt{7595536-1440011520}}{2\times 13158}$$

Tāpiri 7595536 ki te -1440011520.

$$x=\frac{-\left(-2756\right)±\sqrt{-1432415984}}{2\times 13158}$$

Tuhia te pūtakerua o te -1432415984.

$$x=\frac{-\left(-2756\right)±4\sqrt{89525999}i}{2\times 13158}$$

Ko te tauaro o -2756 ko 2756.

$$x=\frac{2756±4\sqrt{89525999}i}{2\times 13158}$$

Whakareatia 2 ki te 13158.

$$x=\frac{2756±4\sqrt{89525999}i}{26316}$$

Nā, me whakaoti te whārite x=\frac{2756±4\sqrt{89525999}i}{26316} ina he tāpiri te ±. Tāpiri 2756 ki te 4i\sqrt{89525999}.

$$x=\frac{2756+4\sqrt{89525999}i}{26316}$$

Whakawehe 2756+4i\sqrt{89525999} ki te 26316.

$$x=\frac{689+\sqrt{89525999}i}{6579}$$

Nā, me whakaoti te whārite x=\frac{2756±4\sqrt{89525999}i}{26316} ina he tango te ±. Tango 4i\sqrt{89525999} mai i 2756.

$$x=\frac{-4\sqrt{89525999}i+2756}{26316}$$

Whakawehe 2756-4i\sqrt{89525999} ki te 26316.

$$x=\frac{-\sqrt{89525999}i+689}{6579}$$

Kua oti te whārite te whakatau.

$$x=\frac{689+\sqrt{89525999}i}{6579} x=\frac{-\sqrt{89525999}i+689}{6579}$$

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.

$$13158x^{2}-2756x+27360=0$$

Me tango 27360 mai i ngā taha e rua o te whārite.

$$13158x^{2}-2756x+27360-27360=-27360$$

Mā te tango i te 27360 i a ia ake anō ka toe ko te 0.

$$13158x^{2}-2756x=-27360$$

Whakawehea ngā taha e rua ki te 13158.

$$\frac{13158x^{2}-2756x}{13158}=-\frac{27360}{13158}$$

Mā te whakawehe ki te 13158 ka wetekia te whakareanga ki te 13158.

$$x^{2}+\left(-\frac{2756}{13158}\right)x=-\frac{27360}{13158}$$

Whakahekea te hautanga \frac{-2756}{13158} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{1378}{6579}x=-\frac{27360}{13158}$$

Whakahekea te hautanga \frac{-27360}{13158} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 18.

$$x^{2}-\frac{1378}{6579}x=-\frac{1520}{731}$$

Whakawehea te -\frac{1378}{6579}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{689}{6579}. Nā, tāpiria te pūrua o te -\frac{689}{6579} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1378}{6579}x+\left(-\frac{689}{6579}\right)^{2}=-\frac{1520}{731}+\left(-\frac{689}{6579}\right)^{2}$$

Pūruatia -\frac{689}{6579} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1378}{6579}x+\frac{474721}{43283241}=-\frac{1520}{731}+\frac{474721}{43283241}$$

Tāpiri -\frac{1520}{731} ki te \frac{474721}{43283241} 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.

$$x^{2}-\frac{1378}{6579}x+\frac{474721}{43283241}=-\frac{89525999}{43283241}$$

Tauwehea x^{2}-\frac{1378}{6579}x+\frac{474721}{43283241}. 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}.

$$\left(x-\frac{689}{6579}\right)^{2}=-\frac{89525999}{43283241}$$

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

$$\sqrt{\left(x-\frac{689}{6579}\right)^{2}}=\sqrt{-\frac{89525999}{43283241}}$$

Whakarūnātia.

$$x-\frac{689}{6579}=\frac{\sqrt{89525999}i}{6579} x-\frac{689}{6579}=-\frac{\sqrt{89525999}i}{6579}$$

Me tāpiri \frac{689}{6579} ki ngā taha e rua o te whārite.

$$x=\frac{689+\sqrt{89525999}i}{6579} x=\frac{-\sqrt{89525999}i+689}{6579}$$