Whakaoti i te z^2-25*10^-12z+16*10^-12=0

$${ z }^{ 2 } -25 \times { 10 }^{ -12 } z+16 \times { 10 }^{ -12 } = 0$$

Whakaoti mō z

$$z=\frac{1+\sqrt{102399999999}i}{80000000000}\approx 1.25 \cdot 10^{-11}+0.000004i$$

$$z=\frac{-\sqrt{102399999999}i+1}{80000000000}\approx 1.25 \cdot 10^{-11}-0.000004i$$

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.

$$z^{2}-\frac{1}{40000000000}z+\frac{1}{62500000000}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -\frac{1}{40000000000} mō b, me \frac{1}{62500000000} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-\frac{1}{40000000000}\right)±\sqrt{\left(-\frac{1}{40000000000}\right)^{2}-4\times \frac{1}{62500000000}}}{2}$$

Pūruatia -\frac{1}{40000000000} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$z=\frac{-\left(-\frac{1}{40000000000}\right)±\sqrt{\frac{1}{1600000000000000000000}-4\times \frac{1}{62500000000}}}{2}$$

Whakareatia -4 ki te \frac{1}{62500000000}.

$$z=\frac{-\left(-\frac{1}{40000000000}\right)±\sqrt{\frac{1}{1600000000000000000000}-\frac{1}{15625000000}}}{2}$$

Tāpiri \frac{1}{1600000000000000000000} ki te -\frac{1}{15625000000} 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.

$$z=\frac{-\left(-\frac{1}{40000000000}\right)±\sqrt{-\frac{102399999999}{1600000000000000000000}}}{2}$$

Tuhia te pūtakerua o te -\frac{102399999999}{1600000000000000000000}.

$$z=\frac{-\left(-\frac{1}{40000000000}\right)±\frac{\sqrt{102399999999}i}{40000000000}}{2}$$

Ko te tauaro o -\frac{1}{40000000000} ko \frac{1}{40000000000}.

$$z=\frac{\frac{1}{40000000000}±\frac{\sqrt{102399999999}i}{40000000000}}{2}$$

Nā, me whakaoti te whārite z=\frac{\frac{1}{40000000000}±\frac{\sqrt{102399999999}i}{40000000000}}{2} ina he tāpiri te ±. Tāpiri \frac{1}{40000000000} ki te \frac{i\sqrt{102399999999}}{40000000000}.

$$z=\frac{1+\sqrt{102399999999}i}{2\times 40000000000}$$

Whakawehe \frac{1+i\sqrt{102399999999}}{40000000000} ki te 2.

$$z=\frac{1+\sqrt{102399999999}i}{80000000000}$$

Nā, me whakaoti te whārite z=\frac{\frac{1}{40000000000}±\frac{\sqrt{102399999999}i}{40000000000}}{2} ina he tango te ±. Tango \frac{i\sqrt{102399999999}}{40000000000} mai i \frac{1}{40000000000}.

$$z=\frac{-\sqrt{102399999999}i+1}{2\times 40000000000}$$

Whakawehe \frac{1-i\sqrt{102399999999}}{40000000000} ki te 2.

$$z=\frac{-\sqrt{102399999999}i+1}{80000000000}$$

Kua oti te whārite te whakatau.

$$z=\frac{1+\sqrt{102399999999}i}{80000000000} z=\frac{-\sqrt{102399999999}i+1}{80000000000}$$

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.

$$z^{2}-\frac{1}{40000000000}z+\frac{1}{62500000000}=0$$

Me tango \frac{1}{62500000000} mai i ngā taha e rua o te whārite.

$$z^{2}-\frac{1}{40000000000}z+\frac{1}{62500000000}-\frac{1}{62500000000}=-\frac{1}{62500000000}$$

Mā te tango i te \frac{1}{62500000000} i a ia ake anō ka toe ko te 0.

$$z^{2}-\frac{1}{40000000000}z=-\frac{1}{62500000000}$$

Whakawehea te -\frac{1}{40000000000}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{80000000000}. Nā, tāpiria te pūrua o te -\frac{1}{80000000000} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$z^{2}-\frac{1}{40000000000}z+\left(-\frac{1}{80000000000}\right)^{2}=-\frac{1}{62500000000}+\left(-\frac{1}{80000000000}\right)^{2}$$

Pūruatia -\frac{1}{80000000000} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$z^{2}-\frac{1}{40000000000}z+\frac{1}{6400000000000000000000}=-\frac{1}{62500000000}+\frac{1}{6400000000000000000000}$$

Tāpiri -\frac{1}{62500000000} ki te \frac{1}{6400000000000000000000} 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.

$$z^{2}-\frac{1}{40000000000}z+\frac{1}{6400000000000000000000}=-\frac{102399999999}{6400000000000000000000}$$

Tauwehea z^{2}-\frac{1}{40000000000}z+\frac{1}{6400000000000000000000}. 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}.

$$\left(z-\frac{1}{80000000000}\right)^{2}=-\frac{102399999999}{6400000000000000000000}$$

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

$$\sqrt{\left(z-\frac{1}{80000000000}\right)^{2}}=\sqrt{-\frac{102399999999}{6400000000000000000000}}$$

Whakarūnātia.

$$z-\frac{1}{80000000000}=\frac{\sqrt{102399999999}i}{80000000000} z-\frac{1}{80000000000}=-\frac{\sqrt{102399999999}i}{80000000000}$$

Me tāpiri \frac{1}{80000000000} ki ngā taha e rua o te whārite.

$$z=\frac{1+\sqrt{102399999999}i}{80000000000} z=\frac{-\sqrt{102399999999}i+1}{80000000000}$$