Whakaoti i te 27=22t-5t^2

$$27 = 22 t - 5 t ^ { 2 }$$

Whakaoti mō t

$$t=\frac{11+\sqrt{14}i}{5}\approx 2.2+0.748331477i$$

$$t=\frac{-\sqrt{14}i+11}{5}\approx 2.2-0.748331477i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$22t-5t^{2}=27$$

Tangohia te 27 mai i ngā taha e rua.

$$22t-5t^{2}-27=0$$

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.

$$-5t^{2}+22t-27=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -5 mō a, 22 mō b, me -27 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-22±\sqrt{22^{2}-4\left(-5\right)\left(-27\right)}}{2\left(-5\right)}$$

Pūrua 22.

$$t=\frac{-22±\sqrt{484-4\left(-5\right)\left(-27\right)}}{2\left(-5\right)}$$

Whakareatia -4 ki te -5.

$$t=\frac{-22±\sqrt{484+20\left(-27\right)}}{2\left(-5\right)}$$

Whakareatia 20 ki te -27.

$$t=\frac{-22±\sqrt{484-540}}{2\left(-5\right)}$$

Tāpiri 484 ki te -540.

$$t=\frac{-22±\sqrt{-56}}{2\left(-5\right)}$$

Tuhia te pūtakerua o te -56.

$$t=\frac{-22±2\sqrt{14}i}{2\left(-5\right)}$$

Whakareatia 2 ki te -5.

$$t=\frac{-22±2\sqrt{14}i}{-10}$$

Nā, me whakaoti te whārite t=\frac{-22±2\sqrt{14}i}{-10} ina he tāpiri te ±. Tāpiri -22 ki te 2i\sqrt{14}.

$$t=\frac{-22+2\sqrt{14}i}{-10}$$

Whakawehe -22+2i\sqrt{14} ki te -10.

$$t=\frac{-\sqrt{14}i+11}{5}$$

Nā, me whakaoti te whārite t=\frac{-22±2\sqrt{14}i}{-10} ina he tango te ±. Tango 2i\sqrt{14} mai i -22.

$$t=\frac{-2\sqrt{14}i-22}{-10}$$

Whakawehe -22-2i\sqrt{14} ki te -10.

$$t=\frac{11+\sqrt{14}i}{5}$$

Kua oti te whārite te whakatau.

$$t=\frac{-\sqrt{14}i+11}{5} t=\frac{11+\sqrt{14}i}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$22t-5t^{2}=27$$

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.

$$-5t^{2}+22t=27$$

Whakawehea ngā taha e rua ki te -5.

$$\frac{-5t^{2}+22t}{-5}=\frac{27}{-5}$$

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

$$t^{2}+\frac{22}{-5}t=\frac{27}{-5}$$

Whakawehe 22 ki te -5.

$$t^{2}-\frac{22}{5}t=\frac{27}{-5}$$

Whakawehe 27 ki te -5.

$$t^{2}-\frac{22}{5}t=-\frac{27}{5}$$

Whakawehea te -\frac{22}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{5}. Nā, tāpiria te pūrua o te -\frac{11}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{22}{5}t+\left(-\frac{11}{5}\right)^{2}=-\frac{27}{5}+\left(-\frac{11}{5}\right)^{2}$$

Pūruatia -\frac{11}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{22}{5}t+\frac{121}{25}=-\frac{27}{5}+\frac{121}{25}$$

Tāpiri -\frac{27}{5} ki te \frac{121}{25} 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.

$$t^{2}-\frac{22}{5}t+\frac{121}{25}=-\frac{14}{25}$$

Tauwehea t^{2}-\frac{22}{5}t+\frac{121}{25}. 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}.

$$\left(t-\frac{11}{5}\right)^{2}=-\frac{14}{25}$$

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

$$\sqrt{\left(t-\frac{11}{5}\right)^{2}}=\sqrt{-\frac{14}{25}}$$

Whakarūnātia.

$$t-\frac{11}{5}=\frac{\sqrt{14}i}{5} t-\frac{11}{5}=-\frac{\sqrt{14}i}{5}$$

Me tāpiri \frac{11}{5} ki ngā taha e rua o te whārite.

$$t=\frac{11+\sqrt{14}i}{5} t=\frac{-\sqrt{14}i+11}{5}$$