Whakaoti i te 100=5t+2t^2

$$100 = 5t+2 { t }^{ 2 }$$

Whakaoti mō t

$$t = \frac{5 \sqrt{33} - 5}{4} \approx 5.930703308$$

$$t=\frac{-5\sqrt{33}-5}{4}\approx -8.430703308$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$5t+2t^{2}=100$$

Tangohia te 100 mai i ngā taha e rua.

$$5t+2t^{2}-100=0$$

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.

$$2t^{2}+5t-100=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 5 mō b, me -100 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-5±\sqrt{5^{2}-4\times 2\left(-100\right)}}{2\times 2}$$

Pūrua 5.

$$t=\frac{-5±\sqrt{25-4\times 2\left(-100\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$t=\frac{-5±\sqrt{25-8\left(-100\right)}}{2\times 2}$$

Whakareatia -8 ki te -100.

$$t=\frac{-5±\sqrt{25+800}}{2\times 2}$$

Tāpiri 25 ki te 800.

$$t=\frac{-5±\sqrt{825}}{2\times 2}$$

Tuhia te pūtakerua o te 825.

$$t=\frac{-5±5\sqrt{33}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$t=\frac{-5±5\sqrt{33}}{4}$$

Nā, me whakaoti te whārite t=\frac{-5±5\sqrt{33}}{4} ina he tāpiri te ±. Tāpiri -5 ki te 5\sqrt{33}.

$$t=\frac{5\sqrt{33}-5}{4}$$

Nā, me whakaoti te whārite t=\frac{-5±5\sqrt{33}}{4} ina he tango te ±. Tango 5\sqrt{33} mai i -5.

$$t=\frac{-5\sqrt{33}-5}{4}$$

Kua oti te whārite te whakatau.

$$t=\frac{5\sqrt{33}-5}{4} t=\frac{-5\sqrt{33}-5}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$5t+2t^{2}=100$$

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.

$$2t^{2}+5t=100$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2t^{2}+5t}{2}=\frac{100}{2}$$

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

$$t^{2}+\frac{5}{2}t=\frac{100}{2}$$

Whakawehe 100 ki te 2.

$$t^{2}+\frac{5}{2}t=50$$

Whakawehea te \frac{5}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{4}. Nā, tāpiria te pūrua o te \frac{5}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}+\frac{5}{2}t+\left(\frac{5}{4}\right)^{2}=50+\left(\frac{5}{4}\right)^{2}$$

Pūruatia \frac{5}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}+\frac{5}{2}t+\frac{25}{16}=50+\frac{25}{16}$$

Tāpiri 50 ki te \frac{25}{16}.

$$t^{2}+\frac{5}{2}t+\frac{25}{16}=\frac{825}{16}$$

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

$$\left(t+\frac{5}{4}\right)^{2}=\frac{825}{16}$$

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

$$\sqrt{\left(t+\frac{5}{4}\right)^{2}}=\sqrt{\frac{825}{16}}$$

Whakarūnātia.

$$t+\frac{5}{4}=\frac{5\sqrt{33}}{4} t+\frac{5}{4}=-\frac{5\sqrt{33}}{4}$$

Me tango \frac{5}{4} mai i ngā taha e rua o te whārite.

$$t=\frac{5\sqrt{33}-5}{4} t=\frac{-5\sqrt{33}-5}{4}$$