Whakaoti i te -16/5t^2+6t=45

$$- \frac { 16 } { 5 } t ^ { 2 } + 6 t = 45$$

Whakaoti mō t

$$t=\frac{15+15\sqrt{15}i}{16}\approx 0.9375+3.630921887i$$

$$t=\frac{-15\sqrt{15}i+15}{16}\approx 0.9375-3.630921887i$$

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.

$$-\frac{16}{5}t^{2}+6t=45$$

Me tango 45 mai i ngā taha e rua o te whārite.

$$-\frac{16}{5}t^{2}+6t-45=45-45$$

Mā te tango i te 45 i a ia ake anō ka toe ko te 0.

$$-\frac{16}{5}t^{2}+6t-45=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -\frac{16}{5} mō a, 6 mō b, me -45 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-6±\sqrt{6^{2}-4\left(-\frac{16}{5}\right)\left(-45\right)}}{2\left(-\frac{16}{5}\right)}$$

Pūrua 6.

$$t=\frac{-6±\sqrt{36-4\left(-\frac{16}{5}\right)\left(-45\right)}}{2\left(-\frac{16}{5}\right)}$$

Whakareatia -4 ki te -\frac{16}{5}.

$$t=\frac{-6±\sqrt{36+\frac{64}{5}\left(-45\right)}}{2\left(-\frac{16}{5}\right)}$$

Whakareatia \frac{64}{5} ki te -45.

$$t=\frac{-6±\sqrt{36-576}}{2\left(-\frac{16}{5}\right)}$$

Tāpiri 36 ki te -576.

$$t=\frac{-6±\sqrt{-540}}{2\left(-\frac{16}{5}\right)}$$

Tuhia te pūtakerua o te -540.

$$t=\frac{-6±6\sqrt{15}i}{2\left(-\frac{16}{5}\right)}$$

Whakareatia 2 ki te -\frac{16}{5}.

$$t=\frac{-6±6\sqrt{15}i}{-\frac{32}{5}}$$

Nā, me whakaoti te whārite t=\frac{-6±6\sqrt{15}i}{-\frac{32}{5}} ina he tāpiri te ±. Tāpiri -6 ki te 6i\sqrt{15}.

$$t=\frac{-6+6\sqrt{15}i}{-\frac{32}{5}}$$

Whakawehe -6+6i\sqrt{15} ki te -\frac{32}{5} mā te whakarea -6+6i\sqrt{15} ki te tau huripoki o -\frac{32}{5}.

$$t=\frac{-15\sqrt{15}i+15}{16}$$

Nā, me whakaoti te whārite t=\frac{-6±6\sqrt{15}i}{-\frac{32}{5}} ina he tango te ±. Tango 6i\sqrt{15} mai i -6.

$$t=\frac{-6\sqrt{15}i-6}{-\frac{32}{5}}$$

Whakawehe -6-6i\sqrt{15} ki te -\frac{32}{5} mā te whakarea -6-6i\sqrt{15} ki te tau huripoki o -\frac{32}{5}.

$$t=\frac{15+15\sqrt{15}i}{16}$$

Kua oti te whārite te whakatau.

$$t=\frac{-15\sqrt{15}i+15}{16} t=\frac{15+15\sqrt{15}i}{16}$$

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.

$$-\frac{16}{5}t^{2}+6t=45$$

Whakawehea ngā taha e rua o te whārite ki te -\frac{16}{5}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{-\frac{16}{5}t^{2}+6t}{-\frac{16}{5}}=\frac{45}{-\frac{16}{5}}$$

Mā te whakawehe ki te -\frac{16}{5} ka wetekia te whakareanga ki te -\frac{16}{5}.

$$t^{2}+\frac{6}{-\frac{16}{5}}t=\frac{45}{-\frac{16}{5}}$$

Whakawehe 6 ki te -\frac{16}{5} mā te whakarea 6 ki te tau huripoki o -\frac{16}{5}.

$$t^{2}-\frac{15}{8}t=\frac{45}{-\frac{16}{5}}$$

Whakawehe 45 ki te -\frac{16}{5} mā te whakarea 45 ki te tau huripoki o -\frac{16}{5}.

$$t^{2}-\frac{15}{8}t=-\frac{225}{16}$$

Whakawehea te -\frac{15}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{15}{16}. Nā, tāpiria te pūrua o te -\frac{15}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{15}{8}t+\left(-\frac{15}{16}\right)^{2}=-\frac{225}{16}+\left(-\frac{15}{16}\right)^{2}$$

Pūruatia -\frac{15}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{15}{8}t+\frac{225}{256}=-\frac{225}{16}+\frac{225}{256}$$

Tāpiri -\frac{225}{16} ki te \frac{225}{256} 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.

$$t^{2}-\frac{15}{8}t+\frac{225}{256}=-\frac{3375}{256}$$

Tauwehea t^{2}-\frac{15}{8}t+\frac{225}{256}. 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}.

$$\left(t-\frac{15}{16}\right)^{2}=-\frac{3375}{256}$$

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

$$\sqrt{\left(t-\frac{15}{16}\right)^{2}}=\sqrt{-\frac{3375}{256}}$$

Whakarūnātia.

$$t-\frac{15}{16}=\frac{15\sqrt{15}i}{16} t-\frac{15}{16}=-\frac{15\sqrt{15}i}{16}$$

Me tāpiri \frac{15}{16} ki ngā taha e rua o te whārite.

$$t=\frac{15+15\sqrt{15}i}{16} t=\frac{-15\sqrt{15}i+15}{16}$$