Whakaoti i te -16t^2+36t+7=0

$$- 16 t ^ { 2 } + 36 t + 7 = 0$$

Whakaoti mō t

$$t = \frac{\sqrt{109} + 9}{8} \approx 2.430038314$$

$$t=\frac{9-\sqrt{109}}{8}\approx -0.180038314$$

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.

$$-16t^{2}+36t+7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -16 mō a, 36 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-36±\sqrt{36^{2}-4\left(-16\right)\times 7}}{2\left(-16\right)}$$

Pūrua 36.

$$t=\frac{-36±\sqrt{1296-4\left(-16\right)\times 7}}{2\left(-16\right)}$$

Whakareatia -4 ki te -16.

$$t=\frac{-36±\sqrt{1296+64\times 7}}{2\left(-16\right)}$$

Whakareatia 64 ki te 7.

$$t=\frac{-36±\sqrt{1296+448}}{2\left(-16\right)}$$

Tāpiri 1296 ki te 448.

$$t=\frac{-36±\sqrt{1744}}{2\left(-16\right)}$$

Tuhia te pūtakerua o te 1744.

$$t=\frac{-36±4\sqrt{109}}{2\left(-16\right)}$$

Whakareatia 2 ki te -16.

$$t=\frac{-36±4\sqrt{109}}{-32}$$

Nā, me whakaoti te whārite t=\frac{-36±4\sqrt{109}}{-32} ina he tāpiri te ±. Tāpiri -36 ki te 4\sqrt{109}.

$$t=\frac{4\sqrt{109}-36}{-32}$$

Whakawehe -36+4\sqrt{109} ki te -32.

$$t=\frac{9-\sqrt{109}}{8}$$

Nā, me whakaoti te whārite t=\frac{-36±4\sqrt{109}}{-32} ina he tango te ±. Tango 4\sqrt{109} mai i -36.

$$t=\frac{-4\sqrt{109}-36}{-32}$$

Whakawehe -36-4\sqrt{109} ki te -32.

$$t=\frac{\sqrt{109}+9}{8}$$

Kua oti te whārite te whakatau.

$$t=\frac{9-\sqrt{109}}{8} t=\frac{\sqrt{109}+9}{8}$$

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.

$$-16t^{2}+36t+7=0$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$-16t^{2}+36t+7-7=-7$$

Mā te tango i te 7 i a ia ake anō ka toe ko te 0.

$$-16t^{2}+36t=-7$$

Whakawehea ngā taha e rua ki te -16.

$$\frac{-16t^{2}+36t}{-16}=-\frac{7}{-16}$$

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

$$t^{2}+\frac{36}{-16}t=-\frac{7}{-16}$$

Whakahekea te hautanga \frac{36}{-16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$t^{2}-\frac{9}{4}t=-\frac{7}{-16}$$

Whakawehe -7 ki te -16.

$$t^{2}-\frac{9}{4}t=\frac{7}{16}$$

Whakawehea te -\frac{9}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{8}. Nā, tāpiria te pūrua o te -\frac{9}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{9}{4}t+\left(-\frac{9}{8}\right)^{2}=\frac{7}{16}+\left(-\frac{9}{8}\right)^{2}$$

Pūruatia -\frac{9}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{9}{4}t+\frac{81}{64}=\frac{7}{16}+\frac{81}{64}$$

Tāpiri \frac{7}{16} ki te \frac{81}{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.

$$t^{2}-\frac{9}{4}t+\frac{81}{64}=\frac{109}{64}$$

Tauwehea t^{2}-\frac{9}{4}t+\frac{81}{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}.

$$\left(t-\frac{9}{8}\right)^{2}=\frac{109}{64}$$

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

$$\sqrt{\left(t-\frac{9}{8}\right)^{2}}=\sqrt{\frac{109}{64}}$$

Whakarūnātia.

$$t-\frac{9}{8}=\frac{\sqrt{109}}{8} t-\frac{9}{8}=-\frac{\sqrt{109}}{8}$$

Me tāpiri \frac{9}{8} ki ngā taha e rua o te whārite.

$$t=\frac{\sqrt{109}+9}{8} t=\frac{9-\sqrt{109}}{8}$$