Whakaoti i te 8x^2-7x+2=0

$$8 { x }^{ 2 } -7x+2=0$$

Whakaoti mō x (complex solution)

$$x=\frac{7+\sqrt{15}i}{16}\approx 0.4375+0.242061459i$$

$$x=\frac{-\sqrt{15}i+7}{16}\approx 0.4375-0.242061459i$$

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.

$$8x^{2}-7x+2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, -7 mō b, me 2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 8\times 2}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-7\right)±\sqrt{49-32\times 2}}{2\times 8}$$

Whakareatia -32 ki te 2.

$$x=\frac{-\left(-7\right)±\sqrt{49-64}}{2\times 8}$$

Tāpiri 49 ki te -64.

$$x=\frac{-\left(-7\right)±\sqrt{-15}}{2\times 8}$$

Tuhia te pūtakerua o te -15.

$$x=\frac{-\left(-7\right)±\sqrt{15}i}{2\times 8}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±\sqrt{15}i}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{7±\sqrt{15}i}{16}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{15}i}{16} ina he tāpiri te ±. Tāpiri 7 ki te i\sqrt{15}.

$$x=\frac{7+\sqrt{15}i}{16}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{15}i}{16} ina he tango te ±. Tango i\sqrt{15} mai i 7.

$$x=\frac{-\sqrt{15}i+7}{16}$$

Kua oti te whārite te whakatau.

$$x=\frac{7+\sqrt{15}i}{16} x=\frac{-\sqrt{15}i+7}{16}$$

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.

$$8x^{2}-7x+2=0$$

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

$$8x^{2}-7x+2-2=-2$$

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

$$8x^{2}-7x=-2$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}-7x}{8}=-\frac{2}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}-\frac{7}{8}x=-\frac{2}{8}$$

Whakahekea te hautanga \frac{-2}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{7}{8}x=-\frac{1}{4}$$

Whakawehea te -\frac{7}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{16}. Nā, tāpiria te pūrua o te -\frac{7}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{8}x+\left(-\frac{7}{16}\right)^{2}=-\frac{1}{4}+\left(-\frac{7}{16}\right)^{2}$$

Pūruatia -\frac{7}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{8}x+\frac{49}{256}=-\frac{1}{4}+\frac{49}{256}$$

Tāpiri -\frac{1}{4} ki te \frac{49}{256} 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.

$$x^{2}-\frac{7}{8}x+\frac{49}{256}=-\frac{15}{256}$$

Tauwehea x^{2}-\frac{7}{8}x+\frac{49}{256}. 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}.

$$\left(x-\frac{7}{16}\right)^{2}=-\frac{15}{256}$$

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

$$\sqrt{\left(x-\frac{7}{16}\right)^{2}}=\sqrt{-\frac{15}{256}}$$

Whakarūnātia.

$$x-\frac{7}{16}=\frac{\sqrt{15}i}{16} x-\frac{7}{16}=-\frac{\sqrt{15}i}{16}$$

Me tāpiri \frac{7}{16} ki ngā taha e rua o te whārite.

$$x=\frac{7+\sqrt{15}i}{16} x=\frac{-\sqrt{15}i+7}{16}$$