Whakaoti i te 57/16t^2-85/16t=-250

$$\frac { 57 } { 16 } t ^ { 2 } - \frac { 85 } { 16 } t = - 250$$

Whakaoti mō t

$$t=\frac{85+5\sqrt{36191}i}{114}\approx 0.745614035+8.343829954i$$

$$t=\frac{-5\sqrt{36191}i+85}{114}\approx 0.745614035-8.343829954i$$

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.

$$\frac{57}{16}t^{2}-\frac{85}{16}t=-250$$

Me tāpiri 250 ki ngā taha e rua o te whārite.

$$\frac{57}{16}t^{2}-\frac{85}{16}t-\left(-250\right)=-250-\left(-250\right)$$

Mā te tango i te -250 i a ia ake anō ka toe ko te 0.

$$\frac{57}{16}t^{2}-\frac{85}{16}t-\left(-250\right)=0$$

Tango -250 mai i 0.

$$\frac{57}{16}t^{2}-\frac{85}{16}t+250=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{57}{16} mō a, -\frac{85}{16} mō b, me 250 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{\left(-\frac{85}{16}\right)^{2}-4\times \frac{57}{16}\times 250}}{2\times \frac{57}{16}}$$

Pūruatia -\frac{85}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{\frac{7225}{256}-4\times \frac{57}{16}\times 250}}{2\times \frac{57}{16}}$$

Whakareatia -4 ki te \frac{57}{16}.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{\frac{7225}{256}-\frac{57}{4}\times 250}}{2\times \frac{57}{16}}$$

Whakareatia -\frac{57}{4} ki te 250.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{\frac{7225}{256}-\frac{7125}{2}}}{2\times \frac{57}{16}}$$

Tāpiri \frac{7225}{256} ki te -\frac{7125}{2} 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.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{-\frac{904775}{256}}}{2\times \frac{57}{16}}$$

Tuhia te pūtakerua o te -\frac{904775}{256}.

$$t=\frac{-\left(-\frac{85}{16}\right)±\frac{5\sqrt{36191}i}{16}}{2\times \frac{57}{16}}$$

Ko te tauaro o -\frac{85}{16} ko \frac{85}{16}.

$$t=\frac{\frac{85}{16}±\frac{5\sqrt{36191}i}{16}}{2\times \frac{57}{16}}$$

Whakareatia 2 ki te \frac{57}{16}.

$$t=\frac{\frac{85}{16}±\frac{5\sqrt{36191}i}{16}}{\frac{57}{8}}$$

Nā, me whakaoti te whārite t=\frac{\frac{85}{16}±\frac{5\sqrt{36191}i}{16}}{\frac{57}{8}} ina he tāpiri te ±. Tāpiri \frac{85}{16} ki te \frac{5i\sqrt{36191}}{16}.

$$t=\frac{85+5\sqrt{36191}i}{\frac{57}{8}\times 16}$$

Whakawehe \frac{85+5i\sqrt{36191}}{16} ki te \frac{57}{8} mā te whakarea \frac{85+5i\sqrt{36191}}{16} ki te tau huripoki o \frac{57}{8}.

$$t=\frac{85+5\sqrt{36191}i}{114}$$

Nā, me whakaoti te whārite t=\frac{\frac{85}{16}±\frac{5\sqrt{36191}i}{16}}{\frac{57}{8}} ina he tango te ±. Tango \frac{5i\sqrt{36191}}{16} mai i \frac{85}{16}.

$$t=\frac{-5\sqrt{36191}i+85}{\frac{57}{8}\times 16}$$

Whakawehe \frac{85-5i\sqrt{36191}}{16} ki te \frac{57}{8} mā te whakarea \frac{85-5i\sqrt{36191}}{16} ki te tau huripoki o \frac{57}{8}.

$$t=\frac{-5\sqrt{36191}i+85}{114}$$

Kua oti te whārite te whakatau.

$$t=\frac{85+5\sqrt{36191}i}{114} t=\frac{-5\sqrt{36191}i+85}{114}$$

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.

$$\frac{57}{16}t^{2}-\frac{85}{16}t=-250$$

Whakawehea ngā taha e rua o te whārite ki te \frac{57}{16}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{\frac{57}{16}t^{2}-\frac{85}{16}t}{\frac{57}{16}}=-\frac{250}{\frac{57}{16}}$$

Mā te whakawehe ki te \frac{57}{16} ka wetekia te whakareanga ki te \frac{57}{16}.

$$t^{2}+\left(-\frac{\frac{85}{16}}{\frac{57}{16}}\right)t=-\frac{250}{\frac{57}{16}}$$

Whakawehe -\frac{85}{16} ki te \frac{57}{16} mā te whakarea -\frac{85}{16} ki te tau huripoki o \frac{57}{16}.

$$t^{2}-\frac{85}{57}t=-\frac{250}{\frac{57}{16}}$$

Whakawehe -250 ki te \frac{57}{16} mā te whakarea -250 ki te tau huripoki o \frac{57}{16}.

$$t^{2}-\frac{85}{57}t=-\frac{4000}{57}$$

Whakawehea te -\frac{85}{57}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{85}{114}. Nā, tāpiria te pūrua o te -\frac{85}{114} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{85}{57}t+\left(-\frac{85}{114}\right)^{2}=-\frac{4000}{57}+\left(-\frac{85}{114}\right)^{2}$$

Pūruatia -\frac{85}{114} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{85}{57}t+\frac{7225}{12996}=-\frac{4000}{57}+\frac{7225}{12996}$$

Tāpiri -\frac{4000}{57} ki te \frac{7225}{12996} 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.

$$t^{2}-\frac{85}{57}t+\frac{7225}{12996}=-\frac{904775}{12996}$$

Tauwehea t^{2}-\frac{85}{57}t+\frac{7225}{12996}. 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}.

$$\left(t-\frac{85}{114}\right)^{2}=-\frac{904775}{12996}$$

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

$$\sqrt{\left(t-\frac{85}{114}\right)^{2}}=\sqrt{-\frac{904775}{12996}}$$

Whakarūnātia.

$$t-\frac{85}{114}=\frac{5\sqrt{36191}i}{114} t-\frac{85}{114}=-\frac{5\sqrt{36191}i}{114}$$

Me tāpiri \frac{85}{114} ki ngā taha e rua o te whārite.

$$t=\frac{85+5\sqrt{36191}i}{114} t=\frac{-5\sqrt{36191}i+85}{114}$$