Whakaoti i te 100=20t+1/2*98t^2

$$100 = 20 t + \frac { 1 } { 2 } \times 98 t ^ { 2 }$$

Whakaoti mō t

$$t = \frac{50 \sqrt{2} - 10}{49} \approx 1.238993431$$

$$t=\frac{-50\sqrt{2}-10}{49}\approx -1.647156696$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te \frac{1}{2} ki te 98, ka 49.

$$100=20t+49t^{2}$$

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

$$20t+49t^{2}=100$$

Tangohia te 100 mai i ngā taha e rua.

$$20t+49t^{2}-100=0$$

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.

$$49t^{2}+20t-100=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 49 mō a, 20 mō b, me -100 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-20±\sqrt{20^{2}-4\times 49\left(-100\right)}}{2\times 49}$$

Pūrua 20.

$$t=\frac{-20±\sqrt{400-4\times 49\left(-100\right)}}{2\times 49}$$

Whakareatia -4 ki te 49.

$$t=\frac{-20±\sqrt{400-196\left(-100\right)}}{2\times 49}$$

Whakareatia -196 ki te -100.

$$t=\frac{-20±\sqrt{400+19600}}{2\times 49}$$

Tāpiri 400 ki te 19600.

$$t=\frac{-20±\sqrt{20000}}{2\times 49}$$

Tuhia te pūtakerua o te 20000.

$$t=\frac{-20±100\sqrt{2}}{2\times 49}$$

Whakareatia 2 ki te 49.

$$t=\frac{-20±100\sqrt{2}}{98}$$

Nā, me whakaoti te whārite t=\frac{-20±100\sqrt{2}}{98} ina he tāpiri te ±. Tāpiri -20 ki te 100\sqrt{2}.

$$t=\frac{100\sqrt{2}-20}{98}$$

Whakawehe -20+100\sqrt{2} ki te 98.

$$t=\frac{50\sqrt{2}-10}{49}$$

Nā, me whakaoti te whārite t=\frac{-20±100\sqrt{2}}{98} ina he tango te ±. Tango 100\sqrt{2} mai i -20.

$$t=\frac{-100\sqrt{2}-20}{98}$$

Whakawehe -20-100\sqrt{2} ki te 98.

$$t=\frac{-50\sqrt{2}-10}{49}$$

Kua oti te whārite te whakatau.

$$t=\frac{50\sqrt{2}-10}{49} t=\frac{-50\sqrt{2}-10}{49}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te \frac{1}{2} ki te 98, ka 49.

$$100=20t+49t^{2}$$

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

$$20t+49t^{2}=100$$

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.

$$49t^{2}+20t=100$$

Whakawehea ngā taha e rua ki te 49.

$$\frac{49t^{2}+20t}{49}=\frac{100}{49}$$

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

$$t^{2}+\frac{20}{49}t=\frac{100}{49}$$

Whakawehea te \frac{20}{49}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{10}{49}. Nā, tāpiria te pūrua o te \frac{10}{49} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}+\frac{20}{49}t+\left(\frac{10}{49}\right)^{2}=\frac{100}{49}+\left(\frac{10}{49}\right)^{2}$$

Pūruatia \frac{10}{49} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}+\frac{20}{49}t+\frac{100}{2401}=\frac{100}{49}+\frac{100}{2401}$$

Tāpiri \frac{100}{49} ki te \frac{100}{2401} 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.

$$t^{2}+\frac{20}{49}t+\frac{100}{2401}=\frac{5000}{2401}$$

Tauwehea t^{2}+\frac{20}{49}t+\frac{100}{2401}. 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}.

$$\left(t+\frac{10}{49}\right)^{2}=\frac{5000}{2401}$$

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

$$\sqrt{\left(t+\frac{10}{49}\right)^{2}}=\sqrt{\frac{5000}{2401}}$$

Whakarūnātia.

$$t+\frac{10}{49}=\frac{50\sqrt{2}}{49} t+\frac{10}{49}=-\frac{50\sqrt{2}}{49}$$

Me tango \frac{10}{49} mai i ngā taha e rua o te whārite.

$$t=\frac{50\sqrt{2}-10}{49} t=\frac{-50\sqrt{2}-10}{49}$$