Whakaoti i te -51t+49t^2=105

$$-51t+49 { t }^{ 2 } = 105$$

Whakaoti mō t

$$t = \frac{\sqrt{23181} + 51}{98} \approx 2.074011008$$

$$t=\frac{51-\sqrt{23181}}{98}\approx -1.033194681$$

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.

$$49t^{2}-51t=105$$

Me tango 105 mai i ngā taha e rua o te whārite.

$$49t^{2}-51t-105=105-105$$

Mā te tango i te 105 i a ia ake anō ka toe ko te 0.

$$49t^{2}-51t-105=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 49 mō a, -51 mō b, me -105 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-51\right)±\sqrt{\left(-51\right)^{2}-4\times 49\left(-105\right)}}{2\times 49}$$

Pūrua -51.

$$t=\frac{-\left(-51\right)±\sqrt{2601-4\times 49\left(-105\right)}}{2\times 49}$$

Whakareatia -4 ki te 49.

$$t=\frac{-\left(-51\right)±\sqrt{2601-196\left(-105\right)}}{2\times 49}$$

Whakareatia -196 ki te -105.

$$t=\frac{-\left(-51\right)±\sqrt{2601+20580}}{2\times 49}$$

Tāpiri 2601 ki te 20580.

$$t=\frac{-\left(-51\right)±\sqrt{23181}}{2\times 49}$$

Ko te tauaro o -51 ko 51.

$$t=\frac{51±\sqrt{23181}}{2\times 49}$$

Whakareatia 2 ki te 49.

$$t=\frac{51±\sqrt{23181}}{98}$$

Nā, me whakaoti te whārite t=\frac{51±\sqrt{23181}}{98} ina he tāpiri te ±. Tāpiri 51 ki te \sqrt{23181}.

$$t=\frac{\sqrt{23181}+51}{98}$$

Nā, me whakaoti te whārite t=\frac{51±\sqrt{23181}}{98} ina he tango te ±. Tango \sqrt{23181} mai i 51.

$$t=\frac{51-\sqrt{23181}}{98}$$

Kua oti te whārite te whakatau.

$$t=\frac{\sqrt{23181}+51}{98} t=\frac{51-\sqrt{23181}}{98}$$

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.

$$49t^{2}-51t=105$$

Whakawehea ngā taha e rua ki te 49.

$$\frac{49t^{2}-51t}{49}=\frac{105}{49}$$

Mā te whakawehe ki te 49 ka wetekia te whakareanga ki te 49.

$$t^{2}-\frac{51}{49}t=\frac{105}{49}$$

Whakahekea te hautanga \frac{105}{49} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 7.

$$t^{2}-\frac{51}{49}t=\frac{15}{7}$$

Whakawehea te -\frac{51}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{51}{98}. Nā, tāpiria te pūrua o te -\frac{51}{98} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{51}{49}t+\left(-\frac{51}{98}\right)^{2}=\frac{15}{7}+\left(-\frac{51}{98}\right)^{2}$$

Pūruatia -\frac{51}{98} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{51}{49}t+\frac{2601}{9604}=\frac{15}{7}+\frac{2601}{9604}$$

Tāpiri \frac{15}{7} ki te \frac{2601}{9604} 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.

$$t^{2}-\frac{51}{49}t+\frac{2601}{9604}=\frac{23181}{9604}$$

Tauwehea t^{2}-\frac{51}{49}t+\frac{2601}{9604}. 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}.

$$\left(t-\frac{51}{98}\right)^{2}=\frac{23181}{9604}$$

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

$$\sqrt{\left(t-\frac{51}{98}\right)^{2}}=\sqrt{\frac{23181}{9604}}$$

Whakarūnātia.

$$t-\frac{51}{98}=\frac{\sqrt{23181}}{98} t-\frac{51}{98}=-\frac{\sqrt{23181}}{98}$$

Me tāpiri \frac{51}{98} ki ngā taha e rua o te whārite.

$$t=\frac{\sqrt{23181}+51}{98} t=\frac{51-\sqrt{23181}}{98}$$