Whakaoti i te 5t^2-72t-108=0

$$5 t ^ { 2 } - 72 t - 108 = 0$$

Whakaoti mō t

$$t = \frac{6 \sqrt{51} + 36}{5} \approx 15.769714114$$

$$t=\frac{36-6\sqrt{51}}{5}\approx -1.369714114$$

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.

$$5t^{2}-72t-108=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -72 mō b, me -108 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 5\left(-108\right)}}{2\times 5}$$

Pūrua -72.

$$t=\frac{-\left(-72\right)±\sqrt{5184-4\times 5\left(-108\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$t=\frac{-\left(-72\right)±\sqrt{5184-20\left(-108\right)}}{2\times 5}$$

Whakareatia -20 ki te -108.

$$t=\frac{-\left(-72\right)±\sqrt{5184+2160}}{2\times 5}$$

Tāpiri 5184 ki te 2160.

$$t=\frac{-\left(-72\right)±\sqrt{7344}}{2\times 5}$$

Tuhia te pūtakerua o te 7344.

$$t=\frac{-\left(-72\right)±12\sqrt{51}}{2\times 5}$$

Ko te tauaro o -72 ko 72.

$$t=\frac{72±12\sqrt{51}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$t=\frac{72±12\sqrt{51}}{10}$$

Nā, me whakaoti te whārite t=\frac{72±12\sqrt{51}}{10} ina he tāpiri te ±. Tāpiri 72 ki te 12\sqrt{51}.

$$t=\frac{12\sqrt{51}+72}{10}$$

Whakawehe 72+12\sqrt{51} ki te 10.

$$t=\frac{6\sqrt{51}+36}{5}$$

Nā, me whakaoti te whārite t=\frac{72±12\sqrt{51}}{10} ina he tango te ±. Tango 12\sqrt{51} mai i 72.

$$t=\frac{72-12\sqrt{51}}{10}$$

Whakawehe 72-12\sqrt{51} ki te 10.

$$t=\frac{36-6\sqrt{51}}{5}$$

Kua oti te whārite te whakatau.

$$t=\frac{6\sqrt{51}+36}{5} t=\frac{36-6\sqrt{51}}{5}$$

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.

$$5t^{2}-72t-108=0$$

Me tāpiri 108 ki ngā taha e rua o te whārite.

$$5t^{2}-72t-108-\left(-108\right)=-\left(-108\right)$$

Mā te tango i te -108 i a ia ake anō ka toe ko te 0.

$$5t^{2}-72t=-\left(-108\right)$$

Tango -108 mai i 0.

$$5t^{2}-72t=108$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5t^{2}-72t}{5}=\frac{108}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$t^{2}-\frac{72}{5}t=\frac{108}{5}$$

Whakawehea te -\frac{72}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{36}{5}. Nā, tāpiria te pūrua o te -\frac{36}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{72}{5}t+\left(-\frac{36}{5}\right)^{2}=\frac{108}{5}+\left(-\frac{36}{5}\right)^{2}$$

Pūruatia -\frac{36}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{72}{5}t+\frac{1296}{25}=\frac{108}{5}+\frac{1296}{25}$$

Tāpiri \frac{108}{5} ki te \frac{1296}{25} 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.

$$t^{2}-\frac{72}{5}t+\frac{1296}{25}=\frac{1836}{25}$$

Tauwehea t^{2}-\frac{72}{5}t+\frac{1296}{25}. 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}.

$$\left(t-\frac{36}{5}\right)^{2}=\frac{1836}{25}$$

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

$$\sqrt{\left(t-\frac{36}{5}\right)^{2}}=\sqrt{\frac{1836}{25}}$$

Whakarūnātia.

$$t-\frac{36}{5}=\frac{6\sqrt{51}}{5} t-\frac{36}{5}=-\frac{6\sqrt{51}}{5}$$

Me tāpiri \frac{36}{5} ki ngā taha e rua o te whārite.

$$t=\frac{6\sqrt{51}+36}{5} t=\frac{36-6\sqrt{51}}{5}$$