Whakaoti i te -16t^2+92t+20=0

$$- 16 t ^ { 2 } + 92 t + 20 = 0$$

Whakaoti mō t

$$t = \frac{\sqrt{609} + 23}{8} \approx 5.95974067$$

$$t=\frac{23-\sqrt{609}}{8}\approx -0.20974067$$

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.

$$-16t^{2}+92t+20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -16 mō a, 92 mō b, me 20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-92±\sqrt{92^{2}-4\left(-16\right)\times 20}}{2\left(-16\right)}$$

Pūrua 92.

$$t=\frac{-92±\sqrt{8464-4\left(-16\right)\times 20}}{2\left(-16\right)}$$

Whakareatia -4 ki te -16.

$$t=\frac{-92±\sqrt{8464+64\times 20}}{2\left(-16\right)}$$

Whakareatia 64 ki te 20.

$$t=\frac{-92±\sqrt{8464+1280}}{2\left(-16\right)}$$

Tāpiri 8464 ki te 1280.

$$t=\frac{-92±\sqrt{9744}}{2\left(-16\right)}$$

Tuhia te pūtakerua o te 9744.

$$t=\frac{-92±4\sqrt{609}}{2\left(-16\right)}$$

Whakareatia 2 ki te -16.

$$t=\frac{-92±4\sqrt{609}}{-32}$$

Nā, me whakaoti te whārite t=\frac{-92±4\sqrt{609}}{-32} ina he tāpiri te ±. Tāpiri -92 ki te 4\sqrt{609}.

$$t=\frac{4\sqrt{609}-92}{-32}$$

Whakawehe -92+4\sqrt{609} ki te -32.

$$t=\frac{23-\sqrt{609}}{8}$$

Nā, me whakaoti te whārite t=\frac{-92±4\sqrt{609}}{-32} ina he tango te ±. Tango 4\sqrt{609} mai i -92.

$$t=\frac{-4\sqrt{609}-92}{-32}$$

Whakawehe -92-4\sqrt{609} ki te -32.

$$t=\frac{\sqrt{609}+23}{8}$$

Kua oti te whārite te whakatau.

$$t=\frac{23-\sqrt{609}}{8} t=\frac{\sqrt{609}+23}{8}$$

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.

$$-16t^{2}+92t+20=0$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$-16t^{2}+92t+20-20=-20$$

Mā te tango i te 20 i a ia ake anō ka toe ko te 0.

$$-16t^{2}+92t=-20$$

Whakawehea ngā taha e rua ki te -16.

$$\frac{-16t^{2}+92t}{-16}=-\frac{20}{-16}$$

Mā te whakawehe ki te -16 ka wetekia te whakareanga ki te -16.

$$t^{2}+\frac{92}{-16}t=-\frac{20}{-16}$$

Whakahekea te hautanga \frac{92}{-16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$t^{2}-\frac{23}{4}t=-\frac{20}{-16}$$

Whakahekea te hautanga \frac{-20}{-16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$t^{2}-\frac{23}{4}t=\frac{5}{4}$$

Whakawehea te -\frac{23}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{23}{8}. Nā, tāpiria te pūrua o te -\frac{23}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{23}{4}t+\left(-\frac{23}{8}\right)^{2}=\frac{5}{4}+\left(-\frac{23}{8}\right)^{2}$$

Pūruatia -\frac{23}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{23}{4}t+\frac{529}{64}=\frac{5}{4}+\frac{529}{64}$$

Tāpiri \frac{5}{4} ki te \frac{529}{64} 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.

$$t^{2}-\frac{23}{4}t+\frac{529}{64}=\frac{609}{64}$$

Tauwehea t^{2}-\frac{23}{4}t+\frac{529}{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}.

$$\left(t-\frac{23}{8}\right)^{2}=\frac{609}{64}$$

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

$$\sqrt{\left(t-\frac{23}{8}\right)^{2}}=\sqrt{\frac{609}{64}}$$

Whakarūnātia.

$$t-\frac{23}{8}=\frac{\sqrt{609}}{8} t-\frac{23}{8}=-\frac{\sqrt{609}}{8}$$

Me tāpiri \frac{23}{8} ki ngā taha e rua o te whārite.

$$t=\frac{\sqrt{609}+23}{8} t=\frac{23-\sqrt{609}}{8}$$