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2z^{2}+3z+2=0
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=\frac{-3±\sqrt{3^{2}-4\times 2\times 2}}{2\times 2}
Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 3 mō b, me 2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
z=\frac{-3±\sqrt{9-4\times 2\times 2}}{2\times 2}
Pūrua 3.
z=\frac{-3±\sqrt{9-8\times 2}}{2\times 2}
Whakareatia -4 ki te 2.
z=\frac{-3±\sqrt{9-16}}{2\times 2}
Whakareatia -8 ki te 2.
z=\frac{-3±\sqrt{-7}}{2\times 2}
Tāpiri 9 ki te -16.
z=\frac{-3±\sqrt{7}i}{2\times 2}
Tuhia te pūtakerua o te -7.
z=\frac{-3±\sqrt{7}i}{4}
Whakareatia 2 ki te 2.
z=\frac{-3+\sqrt{7}i}{4}
Nā, me whakaoti te whārite z=\frac{-3±\sqrt{7}i}{4} ina he tāpiri te ±. Tāpiri -3 ki te i\sqrt{7}.
z=\frac{-\sqrt{7}i-3}{4}
Nā, me whakaoti te whārite z=\frac{-3±\sqrt{7}i}{4} ina he tango te ±. Tango i\sqrt{7} mai i -3.
z=\frac{-3+\sqrt{7}i}{4} z=\frac{-\sqrt{7}i-3}{4}
Kua oti te whārite te whakatau.
2z^{2}+3z+2=0
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.
2z^{2}+3z+2-2=-2
Me tango 2 mai i ngā taha e rua o te whārite.
2z^{2}+3z=-2
Mā te tango i te 2 i a ia ake anō ka toe ko te 0.
\frac{2z^{2}+3z}{2}=-\frac{2}{2}
Whakawehea ngā taha e rua ki te 2.
z^{2}+\frac{3}{2}z=-\frac{2}{2}
Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.
z^{2}+\frac{3}{2}z=-1
Whakawehe -2 ki te 2.
z^{2}+\frac{3}{2}z+\left(\frac{3}{4}\right)^{2}=-1+\left(\frac{3}{4}\right)^{2}
Whakawehea te \frac{3}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{4}. Nā, tāpiria te pūrua o te \frac{3}{4} 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}{2}z+\frac{9}{16}=-1+\frac{9}{16}
Pūruatia \frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.
z^{2}+\frac{3}{2}z+\frac{9}{16}=-\frac{7}{16}
Tāpiri -1 ki te \frac{9}{16}.
\left(z+\frac{3}{4}\right)^{2}=-\frac{7}{16}
Tauwehea z^{2}+\frac{3}{2}z+\frac{9}{16}. 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}.
\sqrt{\left(z+\frac{3}{4}\right)^{2}}=\sqrt{-\frac{7}{16}}
Tuhia te pūtakerua o ngā taha e rua o te whārite.
z+\frac{3}{4}=\frac{\sqrt{7}i}{4} z+\frac{3}{4}=-\frac{\sqrt{7}i}{4}
Whakarūnātia.
z=\frac{-3+\sqrt{7}i}{4} z=\frac{-\sqrt{7}i-3}{4}
Me tango \frac{3}{4} mai i ngā taha e rua o te whārite.