Whakaoti i te 9x^2+3x+9=0

$$9 x ^ { 2 } + 3 x + 9 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-1+\sqrt{35}i}{6}\approx -0.166666667+0.986013297i$$

$$x=\frac{-\sqrt{35}i-1}{6}\approx -0.166666667-0.986013297i$$

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.

$$9x^{2}+3x+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, 3 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 9\times 9}}{2\times 9}$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\times 9\times 9}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-3±\sqrt{9-36\times 9}}{2\times 9}$$

Whakareatia -36 ki te 9.

$$x=\frac{-3±\sqrt{9-324}}{2\times 9}$$

Tāpiri 9 ki te -324.

$$x=\frac{-3±\sqrt{-315}}{2\times 9}$$

Tuhia te pūtakerua o te -315.

$$x=\frac{-3±3\sqrt{35}i}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{-3±3\sqrt{35}i}{18}$$

Nā, me whakaoti te whārite x=\frac{-3±3\sqrt{35}i}{18} ina he tāpiri te ±. Tāpiri -3 ki te 3i\sqrt{35}.

$$x=\frac{-3+3\sqrt{35}i}{18}$$

Whakawehe -3+3i\sqrt{35} ki te 18.

$$x=\frac{-1+\sqrt{35}i}{6}$$

Nā, me whakaoti te whārite x=\frac{-3±3\sqrt{35}i}{18} ina he tango te ±. Tango 3i\sqrt{35} mai i -3.

$$x=\frac{-3\sqrt{35}i-3}{18}$$

Whakawehe -3-3i\sqrt{35} ki te 18.

$$x=\frac{-\sqrt{35}i-1}{6}$$

Kua oti te whārite te whakatau.

$$x=\frac{-1+\sqrt{35}i}{6} x=\frac{-\sqrt{35}i-1}{6}$$

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.

$$9x^{2}+3x+9=0$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$9x^{2}+3x+9-9=-9$$

Mā te tango i te 9 i a ia ake anō ka toe ko te 0.

$$9x^{2}+3x=-9$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}+3x}{9}=-\frac{9}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$x^{2}+\frac{3}{9}x=-\frac{9}{9}$$

Whakahekea te hautanga \frac{3}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{1}{3}x=-\frac{9}{9}$$

Whakawehe -9 ki te 9.

$$x^{2}+\frac{1}{3}x=-1$$

Whakawehea te \frac{1}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{6}. Nā, tāpiria te pūrua o te \frac{1}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=-1+\left(\frac{1}{6}\right)^{2}$$

Pūruatia \frac{1}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-1+\frac{1}{36}$$

Tāpiri -1 ki te \frac{1}{36}.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=-\frac{35}{36}$$

Tauwehea te x^{2}+\frac{1}{3}x+\frac{1}{36}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{6}\right)^{2}=-\frac{35}{36}$$

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

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{-\frac{35}{36}}$$

Whakarūnātia.

$$x+\frac{1}{6}=\frac{\sqrt{35}i}{6} x+\frac{1}{6}=-\frac{\sqrt{35}i}{6}$$

Me tango \frac{1}{6} mai i ngā taha e rua o te whārite.

$$x=\frac{-1+\sqrt{35}i}{6} x=\frac{-\sqrt{35}i-1}{6}$$