Whakaoti i te 3x^2+5x+6=0

$$3 { x }^{ 2 } +5x+6=0$$

Whakaoti mō x (complex solution)

$$x=\frac{-5+\sqrt{47}i}{6}\approx -0.833333333+1.1426091i$$

$$x=\frac{-\sqrt{47}i-5}{6}\approx -0.833333333-1.1426091i$$

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.

$$3x^{2}+5x+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 5 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 3\times 6}}{2\times 3}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\times 3\times 6}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-5±\sqrt{25-12\times 6}}{2\times 3}$$

Whakareatia -12 ki te 6.

$$x=\frac{-5±\sqrt{25-72}}{2\times 3}$$

Tāpiri 25 ki te -72.

$$x=\frac{-5±\sqrt{-47}}{2\times 3}$$

Tuhia te pūtakerua o te -47.

$$x=\frac{-5±\sqrt{47}i}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-5±\sqrt{47}i}{6}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{47}i}{6} ina he tāpiri te ±. Tāpiri -5 ki te i\sqrt{47}.

$$x=\frac{-5+\sqrt{47}i}{6}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{47}i}{6} ina he tango te ±. Tango i\sqrt{47} mai i -5.

$$x=\frac{-\sqrt{47}i-5}{6}$$

Kua oti te whārite te whakatau.

$$x=\frac{-5+\sqrt{47}i}{6} x=\frac{-\sqrt{47}i-5}{6}$$

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.

$$3x^{2}+5x+6=0$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$3x^{2}+5x+6-6=-6$$

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

$$3x^{2}+5x=-6$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+5x}{3}=-\frac{6}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\frac{5}{3}x=-\frac{6}{3}$$

Whakawehe -6 ki te 3.

$$x^{2}+\frac{5}{3}x=-2$$

Whakawehea te \frac{5}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{6}. Nā, tāpiria te pūrua o te \frac{5}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{5}{3}x+\left(\frac{5}{6}\right)^{2}=-2+\left(\frac{5}{6}\right)^{2}$$

Pūruatia \frac{5}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=-2+\frac{25}{36}$$

Tāpiri -2 ki te \frac{25}{36}.

$$x^{2}+\frac{5}{3}x+\frac{25}{36}=-\frac{47}{36}$$

Tauwehea x^{2}+\frac{5}{3}x+\frac{25}{36}. 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}.

$$\left(x+\frac{5}{6}\right)^{2}=-\frac{47}{36}$$

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

$$\sqrt{\left(x+\frac{5}{6}\right)^{2}}=\sqrt{-\frac{47}{36}}$$

Whakarūnātia.

$$x+\frac{5}{6}=\frac{\sqrt{47}i}{6} x+\frac{5}{6}=-\frac{\sqrt{47}i}{6}$$

Me tango \frac{5}{6} mai i ngā taha e rua o te whārite.

$$x=\frac{-5+\sqrt{47}i}{6} x=\frac{-\sqrt{47}i-5}{6}$$