Whakaoti i te 8x^2+13x+10=0

$$8 x ^ { 2 } + 13 x + 10 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-13+\sqrt{151}i}{16}\approx -0.8125+0.768012858i$$

$$x=\frac{-\sqrt{151}i-13}{16}\approx -0.8125-0.768012858i$$

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.

$$8x^{2}+13x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, 13 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\times 8\times 10}}{2\times 8}$$

Pūrua 13.

$$x=\frac{-13±\sqrt{169-4\times 8\times 10}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-13±\sqrt{169-32\times 10}}{2\times 8}$$

Whakareatia -32 ki te 10.

$$x=\frac{-13±\sqrt{169-320}}{2\times 8}$$

Tāpiri 169 ki te -320.

$$x=\frac{-13±\sqrt{-151}}{2\times 8}$$

Tuhia te pūtakerua o te -151.

$$x=\frac{-13±\sqrt{151}i}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{-13±\sqrt{151}i}{16}$$

Nā, me whakaoti te whārite x=\frac{-13±\sqrt{151}i}{16} ina he tāpiri te ±. Tāpiri -13 ki te i\sqrt{151}.

$$x=\frac{-13+\sqrt{151}i}{16}$$

Nā, me whakaoti te whārite x=\frac{-13±\sqrt{151}i}{16} ina he tango te ±. Tango i\sqrt{151} mai i -13.

$$x=\frac{-\sqrt{151}i-13}{16}$$

Kua oti te whārite te whakatau.

$$x=\frac{-13+\sqrt{151}i}{16} x=\frac{-\sqrt{151}i-13}{16}$$

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.

$$8x^{2}+13x+10=0$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$8x^{2}+13x+10-10=-10$$

Mā te tango i te 10 i a ia ake anō ka toe ko te 0.

$$8x^{2}+13x=-10$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}+13x}{8}=-\frac{10}{8}$$

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

$$x^{2}+\frac{13}{8}x=-\frac{10}{8}$$

Whakahekea te hautanga \frac{-10}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{13}{8}x=-\frac{5}{4}$$

Whakawehea te \frac{13}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{13}{16}. Nā, tāpiria te pūrua o te \frac{13}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{13}{8}x+\left(\frac{13}{16}\right)^{2}=-\frac{5}{4}+\left(\frac{13}{16}\right)^{2}$$

Pūruatia \frac{13}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{13}{8}x+\frac{169}{256}=-\frac{5}{4}+\frac{169}{256}$$

Tāpiri -\frac{5}{4} ki te \frac{169}{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.

$$x^{2}+\frac{13}{8}x+\frac{169}{256}=-\frac{151}{256}$$

Tauwehea x^{2}+\frac{13}{8}x+\frac{169}{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}.

$$\left(x+\frac{13}{16}\right)^{2}=-\frac{151}{256}$$

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

$$\sqrt{\left(x+\frac{13}{16}\right)^{2}}=\sqrt{-\frac{151}{256}}$$

Whakarūnātia.

$$x+\frac{13}{16}=\frac{\sqrt{151}i}{16} x+\frac{13}{16}=-\frac{\sqrt{151}i}{16}$$

Me tango \frac{13}{16} mai i ngā taha e rua o te whārite.

$$x=\frac{-13+\sqrt{151}i}{16} x=\frac{-\sqrt{151}i-13}{16}$$