Whakaoti i te 1-3z+275z^2-075z^3=0

$$1-3z+275 { z }^{ 2 } -075 { z }^{ 3 } = 0$$

Whakaoti mō z

$$z=\frac{3+\sqrt{1091}i}{550}\approx 0.005454545+0.060055071i$$

$$z=\frac{-\sqrt{1091}i+3}{550}\approx 0.005454545-0.060055071i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te 0 ki te 75, ka 0.

$$1-3z+275z^{2}-0z^{3}=0$$

Ko te tau i whakarea ki te kore ka hua ko te kore.

$$1-3z+275z^{2}-0=0$$

Whakaraupapatia anō ngā kīanga tau.

$$275z^{2}-3z+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 275 mō a, -3 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 275}}{2\times 275}$$

Pūrua -3.

$$z=\frac{-\left(-3\right)±\sqrt{9-4\times 275}}{2\times 275}$$

Whakareatia -4 ki te 275.

$$z=\frac{-\left(-3\right)±\sqrt{9-1100}}{2\times 275}$$

Tāpiri 9 ki te -1100.

$$z=\frac{-\left(-3\right)±\sqrt{-1091}}{2\times 275}$$

Tuhia te pūtakerua o te -1091.

$$z=\frac{-\left(-3\right)±\sqrt{1091}i}{2\times 275}$$

Ko te tauaro o -3 ko 3.

$$z=\frac{3±\sqrt{1091}i}{2\times 275}$$

Whakareatia 2 ki te 275.

$$z=\frac{3±\sqrt{1091}i}{550}$$

Nā, me whakaoti te whārite z=\frac{3±\sqrt{1091}i}{550} ina he tāpiri te ±. Tāpiri 3 ki te i\sqrt{1091}.

$$z=\frac{3+\sqrt{1091}i}{550}$$

Nā, me whakaoti te whārite z=\frac{3±\sqrt{1091}i}{550} ina he tango te ±. Tango i\sqrt{1091} mai i 3.

$$z=\frac{-\sqrt{1091}i+3}{550}$$

Kua oti te whārite te whakatau.

$$z=\frac{3+\sqrt{1091}i}{550} z=\frac{-\sqrt{1091}i+3}{550}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te 0 ki te 75, ka 0.

$$1-3z+275z^{2}-0z^{3}=0$$

Ko te tau i whakarea ki te kore ka hua ko te kore.

$$1-3z+275z^{2}-0=0$$

Me tāpiri te 0 ki ngā taha e rua.

$$1-3z+275z^{2}=0+0$$

Tāpirihia te 0 ki te 0, ka 0.

$$1-3z+275z^{2}=0$$

Tangohia te 1 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-3z+275z^{2}=-1$$

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.

$$275z^{2}-3z=-1$$

Whakawehea ngā taha e rua ki te 275.

$$\frac{275z^{2}-3z}{275}=-\frac{1}{275}$$

Mā te whakawehe ki te 275 ka wetekia te whakareanga ki te 275.

$$z^{2}-\frac{3}{275}z=-\frac{1}{275}$$

Whakawehea te -\frac{3}{275}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{550}. Nā, tāpiria te pūrua o te -\frac{3}{550} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$z^{2}-\frac{3}{275}z+\left(-\frac{3}{550}\right)^{2}=-\frac{1}{275}+\left(-\frac{3}{550}\right)^{2}$$

Pūruatia -\frac{3}{550} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$z^{2}-\frac{3}{275}z+\frac{9}{302500}=-\frac{1}{275}+\frac{9}{302500}$$

Tāpiri -\frac{1}{275} ki te \frac{9}{302500} 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.

$$z^{2}-\frac{3}{275}z+\frac{9}{302500}=-\frac{1091}{302500}$$

Tauwehea z^{2}-\frac{3}{275}z+\frac{9}{302500}. 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}.

$$\left(z-\frac{3}{550}\right)^{2}=-\frac{1091}{302500}$$

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

$$\sqrt{\left(z-\frac{3}{550}\right)^{2}}=\sqrt{-\frac{1091}{302500}}$$

Whakarūnātia.

$$z-\frac{3}{550}=\frac{\sqrt{1091}i}{550} z-\frac{3}{550}=-\frac{\sqrt{1091}i}{550}$$

Me tāpiri \frac{3}{550} ki ngā taha e rua o te whārite.

$$z=\frac{3+\sqrt{1091}i}{550} z=\frac{-\sqrt{1091}i+3}{550}$$