Whakaoti i te 1/2x^2-5/8x+2=0

$$\frac { 1 } { 2 } x ^ { 2 } - \frac { 5 } { 8 } x + 2 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{5+\sqrt{231}i}{8}\approx 0.625+1.899835519i$$

$$x=\frac{-\sqrt{231}i+5}{8}\approx 0.625-1.899835519i$$

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.

$$\frac{1}{2}x^{2}-\frac{5}{8}x+2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{2} mō a, -\frac{5}{8} mō b, me 2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{5}{8}\right)±\sqrt{\left(-\frac{5}{8}\right)^{2}-4\times \frac{1}{2}\times 2}}{2\times \frac{1}{2}}$$

Pūruatia -\frac{5}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{5}{8}\right)±\sqrt{\frac{25}{64}-4\times \frac{1}{2}\times 2}}{2\times \frac{1}{2}}$$

Whakareatia -4 ki te \frac{1}{2}.

$$x=\frac{-\left(-\frac{5}{8}\right)±\sqrt{\frac{25}{64}-2\times 2}}{2\times \frac{1}{2}}$$

Whakareatia -2 ki te 2.

$$x=\frac{-\left(-\frac{5}{8}\right)±\sqrt{\frac{25}{64}-4}}{2\times \frac{1}{2}}$$

Tāpiri \frac{25}{64} ki te -4.

$$x=\frac{-\left(-\frac{5}{8}\right)±\sqrt{-\frac{231}{64}}}{2\times \frac{1}{2}}$$

Tuhia te pūtakerua o te -\frac{231}{64}.

$$x=\frac{-\left(-\frac{5}{8}\right)±\frac{\sqrt{231}i}{8}}{2\times \frac{1}{2}}$$

Ko te tauaro o -\frac{5}{8} ko \frac{5}{8}.

$$x=\frac{\frac{5}{8}±\frac{\sqrt{231}i}{8}}{2\times \frac{1}{2}}$$

Whakareatia 2 ki te \frac{1}{2}.

$$x=\frac{\frac{5}{8}±\frac{\sqrt{231}i}{8}}{1}$$

Nā, me whakaoti te whārite x=\frac{\frac{5}{8}±\frac{\sqrt{231}i}{8}}{1} ina he tāpiri te ±. Tāpiri \frac{5}{8} ki te \frac{i\sqrt{231}}{8}.

$$x=\frac{5+\sqrt{231}i}{8}$$

Nā, me whakaoti te whārite x=\frac{\frac{5}{8}±\frac{\sqrt{231}i}{8}}{1} ina he tango te ±. Tango \frac{i\sqrt{231}}{8} mai i \frac{5}{8}.

$$x=\frac{-\sqrt{231}i+5}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{5+\sqrt{231}i}{8} x=\frac{-\sqrt{231}i+5}{8}$$

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.

$$\frac{1}{2}x^{2}-\frac{5}{8}x+2=0$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$\frac{1}{2}x^{2}-\frac{5}{8}x+2-2=-2$$

Mā te tango i te 2 i a ia ake anō ka toe ko te 0.

$$\frac{1}{2}x^{2}-\frac{5}{8}x=-2$$

Me whakarea ngā taha e rua ki te 2.

$$\frac{\frac{1}{2}x^{2}-\frac{5}{8}x}{\frac{1}{2}}=-\frac{2}{\frac{1}{2}}$$

Mā te whakawehe ki te \frac{1}{2} ka wetekia te whakareanga ki te \frac{1}{2}.

$$x^{2}+\left(-\frac{\frac{5}{8}}{\frac{1}{2}}\right)x=-\frac{2}{\frac{1}{2}}$$

Whakawehe -\frac{5}{8} ki te \frac{1}{2} mā te whakarea -\frac{5}{8} ki te tau huripoki o \frac{1}{2}.

$$x^{2}-\frac{5}{4}x=-\frac{2}{\frac{1}{2}}$$

Whakawehe -2 ki te \frac{1}{2} mā te whakarea -2 ki te tau huripoki o \frac{1}{2}.

$$x^{2}-\frac{5}{4}x=-4$$

Whakawehea te -\frac{5}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{8}. Nā, tāpiria te pūrua o te -\frac{5}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=-4+\left(-\frac{5}{8}\right)^{2}$$

Pūruatia -\frac{5}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-4+\frac{25}{64}$$

Tāpiri -4 ki te \frac{25}{64}.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-\frac{231}{64}$$

Tauwehea x^{2}-\frac{5}{4}x+\frac{25}{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}.

$$\left(x-\frac{5}{8}\right)^{2}=-\frac{231}{64}$$

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

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{-\frac{231}{64}}$$

Whakarūnātia.

$$x-\frac{5}{8}=\frac{\sqrt{231}i}{8} x-\frac{5}{8}=-\frac{\sqrt{231}i}{8}$$

Me tāpiri \frac{5}{8} ki ngā taha e rua o te whārite.

$$x=\frac{5+\sqrt{231}i}{8} x=\frac{-\sqrt{231}i+5}{8}$$