Whakaoti i te -36.34=11.11t-4.9t^2

$$- 36.34 = 11.11 t - 4.9 t ^ { 2 }$$

Whakaoti mō t

$$t = \frac{\sqrt{8356961} + 1111}{980} \approx 4.083511103$$

$$t=\frac{1111-\sqrt{8356961}}{980}\approx -1.816164164$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$11.11t-4.9t^{2}=-36.34$$

Me tāpiri te 36.34 ki ngā taha e rua.

$$11.11t-4.9t^{2}+36.34=0$$

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.

$$-4.9t^{2}+11.11t+36.34=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4.9 mō a, 11.11 mō b, me 36.34 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-11.11±\sqrt{11.11^{2}-4\left(-4.9\right)\times 36.34}}{2\left(-4.9\right)}$$

Pūruatia 11.11 mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t=\frac{-11.11±\sqrt{123.4321-4\left(-4.9\right)\times 36.34}}{2\left(-4.9\right)}$$

Whakareatia -4 ki te -4.9.

$$t=\frac{-11.11±\sqrt{123.4321+19.6\times 36.34}}{2\left(-4.9\right)}$$

Whakareatia 19.6 ki te 36.34 mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$t=\frac{-11.11±\sqrt{123.4321+712.264}}{2\left(-4.9\right)}$$

Tāpiri 123.4321 ki te 712.264 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.

$$t=\frac{-11.11±\sqrt{835.6961}}{2\left(-4.9\right)}$$

Tuhia te pūtakerua o te 835.6961.

$$t=\frac{-11.11±\frac{\sqrt{8356961}}{100}}{2\left(-4.9\right)}$$

Whakareatia 2 ki te -4.9.

$$t=\frac{-11.11±\frac{\sqrt{8356961}}{100}}{-9.8}$$

Nā, me whakaoti te whārite t=\frac{-11.11±\frac{\sqrt{8356961}}{100}}{-9.8} ina he tāpiri te ±. Tāpiri -11.11 ki te \frac{\sqrt{8356961}}{100}.

$$t=\frac{\sqrt{8356961}-1111}{-9.8\times 100}$$

Whakawehe \frac{-1111+\sqrt{8356961}}{100} ki te -9.8 mā te whakarea \frac{-1111+\sqrt{8356961}}{100} ki te tau huripoki o -9.8.

$$t=\frac{1111-\sqrt{8356961}}{980}$$

Nā, me whakaoti te whārite t=\frac{-11.11±\frac{\sqrt{8356961}}{100}}{-9.8} ina he tango te ±. Tango \frac{\sqrt{8356961}}{100} mai i -11.11.

$$t=\frac{-\sqrt{8356961}-1111}{-9.8\times 100}$$

Whakawehe \frac{-1111-\sqrt{8356961}}{100} ki te -9.8 mā te whakarea \frac{-1111-\sqrt{8356961}}{100} ki te tau huripoki o -9.8.

$$t=\frac{\sqrt{8356961}+1111}{980}$$

Kua oti te whārite te whakatau.

$$t=\frac{1111-\sqrt{8356961}}{980} t=\frac{\sqrt{8356961}+1111}{980}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$11.11t-4.9t^{2}=-36.34$$

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.

$$-4.9t^{2}+11.11t=-36.34$$

Whakawehea ngā taha e rua o te whārite ki te -4.9, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{-4.9t^{2}+11.11t}{-4.9}=-\frac{36.34}{-4.9}$$

Mā te whakawehe ki te -4.9 ka wetekia te whakareanga ki te -4.9.

$$t^{2}+\frac{11.11}{-4.9}t=-\frac{36.34}{-4.9}$$

Whakawehe 11.11 ki te -4.9 mā te whakarea 11.11 ki te tau huripoki o -4.9.

$$t^{2}-\frac{1111}{490}t=-\frac{36.34}{-4.9}$$

Whakawehe -36.34 ki te -4.9 mā te whakarea -36.34 ki te tau huripoki o -4.9.

$$t^{2}-\frac{1111}{490}t=\frac{1817}{245}$$

Whakawehea te -\frac{1111}{490}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1111}{980}. Nā, tāpiria te pūrua o te -\frac{1111}{980} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{1111}{490}t+\left(-\frac{1111}{980}\right)^{2}=\frac{1817}{245}+\left(-\frac{1111}{980}\right)^{2}$$

Pūruatia -\frac{1111}{980} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{1111}{490}t+\frac{1234321}{960400}=\frac{1817}{245}+\frac{1234321}{960400}$$

Tāpiri \frac{1817}{245} ki te \frac{1234321}{960400} 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.

$$t^{2}-\frac{1111}{490}t+\frac{1234321}{960400}=\frac{8356961}{960400}$$

Tauwehea t^{2}-\frac{1111}{490}t+\frac{1234321}{960400}. 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}.

$$\left(t-\frac{1111}{980}\right)^{2}=\frac{8356961}{960400}$$

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

$$\sqrt{\left(t-\frac{1111}{980}\right)^{2}}=\sqrt{\frac{8356961}{960400}}$$

Whakarūnātia.

$$t-\frac{1111}{980}=\frac{\sqrt{8356961}}{980} t-\frac{1111}{980}=-\frac{\sqrt{8356961}}{980}$$

Me tāpiri \frac{1111}{980} ki ngā taha e rua o te whārite.

$$t=\frac{\sqrt{8356961}+1111}{980} t=\frac{1111-\sqrt{8356961}}{980}$$