Whakaoti i te 7t^2-32t+12=0

$$7 t ^ { 2 } - 32 t + 12 = 0$$

Whakaoti mō t

$$t = \frac{2 \sqrt{43} + 16}{7} \approx 4.15926815$$

$$t=\frac{16-2\sqrt{43}}{7}\approx 0.412160422$$

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.

$$7t^{2}-32t+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, -32 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 7\times 12}}{2\times 7}$$

Pūrua -32.

$$t=\frac{-\left(-32\right)±\sqrt{1024-4\times 7\times 12}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$t=\frac{-\left(-32\right)±\sqrt{1024-28\times 12}}{2\times 7}$$

Whakareatia -28 ki te 12.

$$t=\frac{-\left(-32\right)±\sqrt{1024-336}}{2\times 7}$$

Tāpiri 1024 ki te -336.

$$t=\frac{-\left(-32\right)±\sqrt{688}}{2\times 7}$$

Tuhia te pūtakerua o te 688.

$$t=\frac{-\left(-32\right)±4\sqrt{43}}{2\times 7}$$

Ko te tauaro o -32 ko 32.

$$t=\frac{32±4\sqrt{43}}{2\times 7}$$

Whakareatia 2 ki te 7.

$$t=\frac{32±4\sqrt{43}}{14}$$

Nā, me whakaoti te whārite t=\frac{32±4\sqrt{43}}{14} ina he tāpiri te ±. Tāpiri 32 ki te 4\sqrt{43}.

$$t=\frac{4\sqrt{43}+32}{14}$$

Whakawehe 32+4\sqrt{43} ki te 14.

$$t=\frac{2\sqrt{43}+16}{7}$$

Nā, me whakaoti te whārite t=\frac{32±4\sqrt{43}}{14} ina he tango te ±. Tango 4\sqrt{43} mai i 32.

$$t=\frac{32-4\sqrt{43}}{14}$$

Whakawehe 32-4\sqrt{43} ki te 14.

$$t=\frac{16-2\sqrt{43}}{7}$$

Kua oti te whārite te whakatau.

$$t=\frac{2\sqrt{43}+16}{7} t=\frac{16-2\sqrt{43}}{7}$$

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.

$$7t^{2}-32t+12=0$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$7t^{2}-32t+12-12=-12$$

Mā te tango i te 12 i a ia ake anō ka toe ko te 0.

$$7t^{2}-32t=-12$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7t^{2}-32t}{7}=-\frac{12}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$t^{2}-\frac{32}{7}t=-\frac{12}{7}$$

Whakawehea te -\frac{32}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{16}{7}. Nā, tāpiria te pūrua o te -\frac{16}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{32}{7}t+\left(-\frac{16}{7}\right)^{2}=-\frac{12}{7}+\left(-\frac{16}{7}\right)^{2}$$

Pūruatia -\frac{16}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{32}{7}t+\frac{256}{49}=-\frac{12}{7}+\frac{256}{49}$$

Tāpiri -\frac{12}{7} ki te \frac{256}{49} 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.

$$t^{2}-\frac{32}{7}t+\frac{256}{49}=\frac{172}{49}$$

Tauwehea t^{2}-\frac{32}{7}t+\frac{256}{49}. 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}.

$$\left(t-\frac{16}{7}\right)^{2}=\frac{172}{49}$$

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

$$\sqrt{\left(t-\frac{16}{7}\right)^{2}}=\sqrt{\frac{172}{49}}$$

Whakarūnātia.

$$t-\frac{16}{7}=\frac{2\sqrt{43}}{7} t-\frac{16}{7}=-\frac{2\sqrt{43}}{7}$$

Me tāpiri \frac{16}{7} ki ngā taha e rua o te whārite.

$$t=\frac{2\sqrt{43}+16}{7} t=\frac{16-2\sqrt{43}}{7}$$