Whakaoti i te 2t^2-7t-7=0

$$2 t ^ { 2 } - 7 t - 7 = 0$$

Whakaoti mō t

$$t = \frac{\sqrt{105} + 7}{4} \approx 4.311737691$$

$$t=\frac{7-\sqrt{105}}{4}\approx -0.811737691$$

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.

$$2t^{2}-7t-7=0$$

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

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

Pūrua -7.

$$t=\frac{-\left(-7\right)±\sqrt{49-4\times 2\left(-7\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$t=\frac{-\left(-7\right)±\sqrt{49-8\left(-7\right)}}{2\times 2}$$

Whakareatia -8 ki te -7.

$$t=\frac{-\left(-7\right)±\sqrt{49+56}}{2\times 2}$$

Tāpiri 49 ki te 56.

$$t=\frac{-\left(-7\right)±\sqrt{105}}{2\times 2}$$

Ko te tauaro o -7 ko 7.

$$t=\frac{7±\sqrt{105}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$t=\frac{7±\sqrt{105}}{4}$$

Nā, me whakaoti te whārite t=\frac{7±\sqrt{105}}{4} ina he tāpiri te ±. Tāpiri 7 ki te \sqrt{105}.

$$t=\frac{\sqrt{105}+7}{4}$$

Nā, me whakaoti te whārite t=\frac{7±\sqrt{105}}{4} ina he tango te ±. Tango \sqrt{105} mai i 7.

$$t=\frac{7-\sqrt{105}}{4}$$

Kua oti te whārite te whakatau.

$$t=\frac{\sqrt{105}+7}{4} t=\frac{7-\sqrt{105}}{4}$$

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.

$$2t^{2}-7t-7=0$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$2t^{2}-7t-7-\left(-7\right)=-\left(-7\right)$$

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

$$2t^{2}-7t=-\left(-7\right)$$

Tango -7 mai i 0.

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

Whakawehea ngā taha e rua ki te 2.

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

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

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

Whakawehea te -\frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{4}. Nā, tāpiria te pūrua o te -\frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{7}{2}t+\frac{49}{16}=\frac{7}{2}+\frac{49}{16}$$

Tāpiri \frac{7}{2} ki te \frac{49}{16} 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.

$$t^{2}-\frac{7}{2}t+\frac{49}{16}=\frac{105}{16}$$

Tauwehea t^{2}-\frac{7}{2}t+\frac{49}{16}. 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}.

$$\left(t-\frac{7}{4}\right)^{2}=\frac{105}{16}$$

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

$$\sqrt{\left(t-\frac{7}{4}\right)^{2}}=\sqrt{\frac{105}{16}}$$

Whakarūnātia.

$$t-\frac{7}{4}=\frac{\sqrt{105}}{4} t-\frac{7}{4}=-\frac{\sqrt{105}}{4}$$

Me tāpiri \frac{7}{4} ki ngā taha e rua o te whārite.

$$t=\frac{\sqrt{105}+7}{4} t=\frac{7-\sqrt{105}}{4}$$