Whakaoti i te 14x^2+9x+10=0

$$14 x ^ { 2 } + 9 x + 10 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-9+\sqrt{479}i}{28}\approx -0.321428571+0.781645308i$$

$$x=\frac{-\sqrt{479}i-9}{28}\approx -0.321428571-0.781645308i$$

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.

$$14x^{2}+9x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 14 mō a, 9 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\times 14\times 10}}{2\times 14}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\times 14\times 10}}{2\times 14}$$

Whakareatia -4 ki te 14.

$$x=\frac{-9±\sqrt{81-56\times 10}}{2\times 14}$$

Whakareatia -56 ki te 10.

$$x=\frac{-9±\sqrt{81-560}}{2\times 14}$$

Tāpiri 81 ki te -560.

$$x=\frac{-9±\sqrt{-479}}{2\times 14}$$

Tuhia te pūtakerua o te -479.

$$x=\frac{-9±\sqrt{479}i}{2\times 14}$$

Whakareatia 2 ki te 14.

$$x=\frac{-9±\sqrt{479}i}{28}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{479}i}{28} ina he tāpiri te ±. Tāpiri -9 ki te i\sqrt{479}.

$$x=\frac{-9+\sqrt{479}i}{28}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{479}i}{28} ina he tango te ±. Tango i\sqrt{479} mai i -9.

$$x=\frac{-\sqrt{479}i-9}{28}$$

Kua oti te whārite te whakatau.

$$x=\frac{-9+\sqrt{479}i}{28} x=\frac{-\sqrt{479}i-9}{28}$$

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.

$$14x^{2}+9x+10=0$$

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

$$14x^{2}+9x+10-10=-10$$

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

$$14x^{2}+9x=-10$$

Whakawehea ngā taha e rua ki te 14.

$$\frac{14x^{2}+9x}{14}=-\frac{10}{14}$$

Mā te whakawehe ki te 14 ka wetekia te whakareanga ki te 14.

$$x^{2}+\frac{9}{14}x=-\frac{10}{14}$$

Whakahekea te hautanga \frac{-10}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{9}{14}x=-\frac{5}{7}$$

Whakawehea te \frac{9}{14}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{9}{28}. Nā, tāpiria te pūrua o te \frac{9}{28} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{9}{14}x+\left(\frac{9}{28}\right)^{2}=-\frac{5}{7}+\left(\frac{9}{28}\right)^{2}$$

Pūruatia \frac{9}{28} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{9}{14}x+\frac{81}{784}=-\frac{5}{7}+\frac{81}{784}$$

Tāpiri -\frac{5}{7} ki te \frac{81}{784} 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.

$$x^{2}+\frac{9}{14}x+\frac{81}{784}=-\frac{479}{784}$$

Tauwehea x^{2}+\frac{9}{14}x+\frac{81}{784}. 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}.

$$\left(x+\frac{9}{28}\right)^{2}=-\frac{479}{784}$$

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

$$\sqrt{\left(x+\frac{9}{28}\right)^{2}}=\sqrt{-\frac{479}{784}}$$

Whakarūnātia.

$$x+\frac{9}{28}=\frac{\sqrt{479}i}{28} x+\frac{9}{28}=-\frac{\sqrt{479}i}{28}$$

Me tango \frac{9}{28} mai i ngā taha e rua o te whārite.

$$x=\frac{-9+\sqrt{479}i}{28} x=\frac{-\sqrt{479}i-9}{28}$$