Whakaoti i te 4t^2-57t+64=0

$$4 t ^ { 2 } - 57 t + 64 = 0$$

Whakaoti mō t

$$t = \frac{5 \sqrt{89} + 57}{8} \approx 13.021238208$$

$$t = \frac{57 - 5 \sqrt{89}}{8} \approx 1.228761792$$

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.

$$4t^{2}-57t+64=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -57 mō b, me 64 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-57\right)±\sqrt{\left(-57\right)^{2}-4\times 4\times 64}}{2\times 4}$$

Pūrua -57.

$$t=\frac{-\left(-57\right)±\sqrt{3249-4\times 4\times 64}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$t=\frac{-\left(-57\right)±\sqrt{3249-16\times 64}}{2\times 4}$$

Whakareatia -16 ki te 64.

$$t=\frac{-\left(-57\right)±\sqrt{3249-1024}}{2\times 4}$$

Tāpiri 3249 ki te -1024.

$$t=\frac{-\left(-57\right)±\sqrt{2225}}{2\times 4}$$

Tuhia te pūtakerua o te 2225.

$$t=\frac{-\left(-57\right)±5\sqrt{89}}{2\times 4}$$

Ko te tauaro o -57 ko 57.

$$t=\frac{57±5\sqrt{89}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$t=\frac{57±5\sqrt{89}}{8}$$

Nā, me whakaoti te whārite t=\frac{57±5\sqrt{89}}{8} ina he tāpiri te ±. Tāpiri 57 ki te 5\sqrt{89}.

$$t=\frac{5\sqrt{89}+57}{8}$$

Nā, me whakaoti te whārite t=\frac{57±5\sqrt{89}}{8} ina he tango te ±. Tango 5\sqrt{89} mai i 57.

$$t=\frac{57-5\sqrt{89}}{8}$$

Kua oti te whārite te whakatau.

$$t=\frac{5\sqrt{89}+57}{8} t=\frac{57-5\sqrt{89}}{8}$$

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.

$$4t^{2}-57t+64=0$$

Me tango 64 mai i ngā taha e rua o te whārite.

$$4t^{2}-57t+64-64=-64$$

Mā te tango i te 64 i a ia ake anō ka toe ko te 0.

$$4t^{2}-57t=-64$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4t^{2}-57t}{4}=-\frac{64}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$t^{2}-\frac{57}{4}t=-\frac{64}{4}$$

Whakawehe -64 ki te 4.

$$t^{2}-\frac{57}{4}t=-16$$

Whakawehea te -\frac{57}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{57}{8}. Nā, tāpiria te pūrua o te -\frac{57}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{57}{4}t+\left(-\frac{57}{8}\right)^{2}=-16+\left(-\frac{57}{8}\right)^{2}$$

Pūruatia -\frac{57}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{57}{4}t+\frac{3249}{64}=-16+\frac{3249}{64}$$

Tāpiri -16 ki te \frac{3249}{64}.

$$t^{2}-\frac{57}{4}t+\frac{3249}{64}=\frac{2225}{64}$$

Tauwehea t^{2}-\frac{57}{4}t+\frac{3249}{64}. 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}.

$$\left(t-\frac{57}{8}\right)^{2}=\frac{2225}{64}$$

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

$$\sqrt{\left(t-\frac{57}{8}\right)^{2}}=\sqrt{\frac{2225}{64}}$$

Whakarūnātia.

$$t-\frac{57}{8}=\frac{5\sqrt{89}}{8} t-\frac{57}{8}=-\frac{5\sqrt{89}}{8}$$

Me tāpiri \frac{57}{8} ki ngā taha e rua o te whārite.

$$t=\frac{5\sqrt{89}+57}{8} t=\frac{57-5\sqrt{89}}{8}$$