Whakaoti i te 5t^2-9t+15=0

$$5 t ^ { 2 } - 9 t + 15 = 0$$

Whakaoti mō t

$$t=\frac{9+\sqrt{219}i}{10}\approx 0.9+1.479864859i$$

$$t=\frac{-\sqrt{219}i+9}{10}\approx 0.9-1.479864859i$$

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.

$$5t^{2}-9t+15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -9 mō b, me 15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 5\times 15}}{2\times 5}$$

Pūrua -9.

$$t=\frac{-\left(-9\right)±\sqrt{81-4\times 5\times 15}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$t=\frac{-\left(-9\right)±\sqrt{81-20\times 15}}{2\times 5}$$

Whakareatia -20 ki te 15.

$$t=\frac{-\left(-9\right)±\sqrt{81-300}}{2\times 5}$$

Tāpiri 81 ki te -300.

$$t=\frac{-\left(-9\right)±\sqrt{-219}}{2\times 5}$$

Tuhia te pūtakerua o te -219.

$$t=\frac{-\left(-9\right)±\sqrt{219}i}{2\times 5}$$

Ko te tauaro o -9 ko 9.

$$t=\frac{9±\sqrt{219}i}{2\times 5}$$

Whakareatia 2 ki te 5.

$$t=\frac{9±\sqrt{219}i}{10}$$

Nā, me whakaoti te whārite t=\frac{9±\sqrt{219}i}{10} ina he tāpiri te ±. Tāpiri 9 ki te i\sqrt{219}.

$$t=\frac{9+\sqrt{219}i}{10}$$

Nā, me whakaoti te whārite t=\frac{9±\sqrt{219}i}{10} ina he tango te ±. Tango i\sqrt{219} mai i 9.

$$t=\frac{-\sqrt{219}i+9}{10}$$

Kua oti te whārite te whakatau.

$$t=\frac{9+\sqrt{219}i}{10} t=\frac{-\sqrt{219}i+9}{10}$$

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.

$$5t^{2}-9t+15=0$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$5t^{2}-9t+15-15=-15$$

Mā te tango i te 15 i a ia ake anō ka toe ko te 0.

$$5t^{2}-9t=-15$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5t^{2}-9t}{5}=-\frac{15}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$t^{2}-\frac{9}{5}t=-\frac{15}{5}$$

Whakawehe -15 ki te 5.

$$t^{2}-\frac{9}{5}t=-3$$

Whakawehea te -\frac{9}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{10}. Nā, tāpiria te pūrua o te -\frac{9}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{9}{5}t+\left(-\frac{9}{10}\right)^{2}=-3+\left(-\frac{9}{10}\right)^{2}$$

Pūruatia -\frac{9}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{9}{5}t+\frac{81}{100}=-3+\frac{81}{100}$$

Tāpiri -3 ki te \frac{81}{100}.

$$t^{2}-\frac{9}{5}t+\frac{81}{100}=-\frac{219}{100}$$

Tauwehea t^{2}-\frac{9}{5}t+\frac{81}{100}. 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}.

$$\left(t-\frac{9}{10}\right)^{2}=-\frac{219}{100}$$

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

$$\sqrt{\left(t-\frac{9}{10}\right)^{2}}=\sqrt{-\frac{219}{100}}$$

Whakarūnātia.

$$t-\frac{9}{10}=\frac{\sqrt{219}i}{10} t-\frac{9}{10}=-\frac{\sqrt{219}i}{10}$$

Me tāpiri \frac{9}{10} ki ngā taha e rua o te whārite.

$$t=\frac{9+\sqrt{219}i}{10} t=\frac{-\sqrt{219}i+9}{10}$$