Whakaoti i te 2x^2-3/2x+7/10=0

$$2 x ^ { 2 } - \frac { 3 } { 2 } x + \frac { 7 } { 10 } = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{\sqrt{335}i}{40}+\frac{3}{8}\approx 0.375+0.45757513i$$

$$x=-\frac{\sqrt{335}i}{40}+\frac{3}{8}\approx 0.375-0.45757513i$$

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.

$$2x^{2}-\frac{3}{2}x+\frac{7}{10}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -\frac{3}{2} mō b, me \frac{7}{10} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\left(-\frac{3}{2}\right)^{2}-4\times 2\times \frac{7}{10}}}{2\times 2}$$

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}-4\times 2\times \frac{7}{10}}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}-8\times \frac{7}{10}}}{2\times 2}$$

Whakareatia -8 ki te \frac{7}{10}.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}-\frac{28}{5}}}{2\times 2}$$

Tāpiri \frac{9}{4} ki te -\frac{28}{5} 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.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{-\frac{67}{20}}}{2\times 2}$$

Tuhia te pūtakerua o te -\frac{67}{20}.

$$x=\frac{-\left(-\frac{3}{2}\right)±\frac{\sqrt{335}i}{10}}{2\times 2}$$

Ko te tauaro o -\frac{3}{2} ko \frac{3}{2}.

$$x=\frac{\frac{3}{2}±\frac{\sqrt{335}i}{10}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{\frac{3}{2}±\frac{\sqrt{335}i}{10}}{4}$$

Nā, me whakaoti te whārite x=\frac{\frac{3}{2}±\frac{\sqrt{335}i}{10}}{4} ina he tāpiri te ±. Tāpiri \frac{3}{2} ki te \frac{i\sqrt{335}}{10}.

$$x=\frac{\frac{\sqrt{335}i}{10}+\frac{3}{2}}{4}$$

Whakawehe \frac{3}{2}+\frac{i\sqrt{335}}{10} ki te 4.

$$x=\frac{\sqrt{335}i}{40}+\frac{3}{8}$$

Nā, me whakaoti te whārite x=\frac{\frac{3}{2}±\frac{\sqrt{335}i}{10}}{4} ina he tango te ±. Tango \frac{i\sqrt{335}}{10} mai i \frac{3}{2}.

$$x=\frac{-\frac{\sqrt{335}i}{10}+\frac{3}{2}}{4}$$

Whakawehe \frac{3}{2}-\frac{i\sqrt{335}}{10} ki te 4.

$$x=-\frac{\sqrt{335}i}{40}+\frac{3}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{335}i}{40}+\frac{3}{8} x=-\frac{\sqrt{335}i}{40}+\frac{3}{8}$$

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.

$$2x^{2}-\frac{3}{2}x+\frac{7}{10}=0$$

Me tango \frac{7}{10} mai i ngā taha e rua o te whārite.

$$2x^{2}-\frac{3}{2}x+\frac{7}{10}-\frac{7}{10}=-\frac{7}{10}$$

Mā te tango i te \frac{7}{10} i a ia ake anō ka toe ko te 0.

$$2x^{2}-\frac{3}{2}x=-\frac{7}{10}$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-\frac{3}{2}x}{2}=-\frac{\frac{7}{10}}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\left(-\frac{\frac{3}{2}}{2}\right)x=-\frac{\frac{7}{10}}{2}$$

Whakawehe -\frac{3}{2} ki te 2.

$$x^{2}-\frac{3}{4}x=-\frac{\frac{7}{10}}{2}$$

Whakawehe -\frac{7}{10} ki te 2.

$$x^{2}-\frac{3}{4}x=-\frac{7}{20}$$

Whakawehea te -\frac{3}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{8}. Nā, tāpiria te pūrua o te -\frac{3}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{4}x+\left(-\frac{3}{8}\right)^{2}=-\frac{7}{20}+\left(-\frac{3}{8}\right)^{2}$$

Pūruatia -\frac{3}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=-\frac{7}{20}+\frac{9}{64}$$

Tāpiri -\frac{7}{20} ki te \frac{9}{64} 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.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=-\frac{67}{320}$$

Tauwehea x^{2}-\frac{3}{4}x+\frac{9}{64}. 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}.

$$\left(x-\frac{3}{8}\right)^{2}=-\frac{67}{320}$$

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

$$\sqrt{\left(x-\frac{3}{8}\right)^{2}}=\sqrt{-\frac{67}{320}}$$

Whakarūnātia.

$$x-\frac{3}{8}=\frac{\sqrt{335}i}{40} x-\frac{3}{8}=-\frac{\sqrt{335}i}{40}$$

Me tāpiri \frac{3}{8} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{335}i}{40}+\frac{3}{8} x=-\frac{\sqrt{335}i}{40}+\frac{3}{8}$$