Whakaoti i te 3t^2-10t-20=0

$$3 t ^ { 2 } - 10 t - 20 = 0$$

Whakaoti mō t

$$t = \frac{\sqrt{85} + 5}{3} \approx 4.739848152$$

$$t=\frac{5-\sqrt{85}}{3}\approx -1.406514819$$

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.

$$3t^{2}-10t-20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -10 mō b, me -20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 3\left(-20\right)}}{2\times 3}$$

Pūrua -10.

$$t=\frac{-\left(-10\right)±\sqrt{100-4\times 3\left(-20\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$t=\frac{-\left(-10\right)±\sqrt{100-12\left(-20\right)}}{2\times 3}$$

Whakareatia -12 ki te -20.

$$t=\frac{-\left(-10\right)±\sqrt{100+240}}{2\times 3}$$

Tāpiri 100 ki te 240.

$$t=\frac{-\left(-10\right)±\sqrt{340}}{2\times 3}$$

Tuhia te pūtakerua o te 340.

$$t=\frac{-\left(-10\right)±2\sqrt{85}}{2\times 3}$$

Ko te tauaro o -10 ko 10.

$$t=\frac{10±2\sqrt{85}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$t=\frac{10±2\sqrt{85}}{6}$$

Nā, me whakaoti te whārite t=\frac{10±2\sqrt{85}}{6} ina he tāpiri te ±. Tāpiri 10 ki te 2\sqrt{85}.

$$t=\frac{2\sqrt{85}+10}{6}$$

Whakawehe 10+2\sqrt{85} ki te 6.

$$t=\frac{\sqrt{85}+5}{3}$$

Nā, me whakaoti te whārite t=\frac{10±2\sqrt{85}}{6} ina he tango te ±. Tango 2\sqrt{85} mai i 10.

$$t=\frac{10-2\sqrt{85}}{6}$$

Whakawehe 10-2\sqrt{85} ki te 6.

$$t=\frac{5-\sqrt{85}}{3}$$

Kua oti te whārite te whakatau.

$$t=\frac{\sqrt{85}+5}{3} t=\frac{5-\sqrt{85}}{3}$$

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.

$$3t^{2}-10t-20=0$$

Me tāpiri 20 ki ngā taha e rua o te whārite.

$$3t^{2}-10t-20-\left(-20\right)=-\left(-20\right)$$

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

$$3t^{2}-10t=-\left(-20\right)$$

Tango -20 mai i 0.

$$3t^{2}-10t=20$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3t^{2}-10t}{3}=\frac{20}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$t^{2}-\frac{10}{3}t=\frac{20}{3}$$

Whakawehea te -\frac{10}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{3}. Nā, tāpiria te pūrua o te -\frac{5}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{10}{3}t+\left(-\frac{5}{3}\right)^{2}=\frac{20}{3}+\left(-\frac{5}{3}\right)^{2}$$

Pūruatia -\frac{5}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{10}{3}t+\frac{25}{9}=\frac{20}{3}+\frac{25}{9}$$

Tāpiri \frac{20}{3} ki te \frac{25}{9} 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.

$$t^{2}-\frac{10}{3}t+\frac{25}{9}=\frac{85}{9}$$

Tauwehea t^{2}-\frac{10}{3}t+\frac{25}{9}. 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}.

$$\left(t-\frac{5}{3}\right)^{2}=\frac{85}{9}$$

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

$$\sqrt{\left(t-\frac{5}{3}\right)^{2}}=\sqrt{\frac{85}{9}}$$

Whakarūnātia.

$$t-\frac{5}{3}=\frac{\sqrt{85}}{3} t-\frac{5}{3}=-\frac{\sqrt{85}}{3}$$

Me tāpiri \frac{5}{3} ki ngā taha e rua o te whārite.

$$t=\frac{\sqrt{85}+5}{3} t=\frac{5-\sqrt{85}}{3}$$