Whakaoti i te -49t^2+98t+100=0

$$- 49 t ^ { 2 } + 98 t + 100 = 0$$

Whakaoti mō t

$$t=\frac{\sqrt{149}}{7}+1\approx 2.743793659$$

$$t=-\frac{\sqrt{149}}{7}+1\approx -0.743793659$$

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.

$$-49t^{2}+98t+100=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -49 mō a, 98 mō b, me 100 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-98±\sqrt{98^{2}-4\left(-49\right)\times 100}}{2\left(-49\right)}$$

Pūrua 98.

$$t=\frac{-98±\sqrt{9604-4\left(-49\right)\times 100}}{2\left(-49\right)}$$

Whakareatia -4 ki te -49.

$$t=\frac{-98±\sqrt{9604+196\times 100}}{2\left(-49\right)}$$

Whakareatia 196 ki te 100.

$$t=\frac{-98±\sqrt{9604+19600}}{2\left(-49\right)}$$

Tāpiri 9604 ki te 19600.

$$t=\frac{-98±\sqrt{29204}}{2\left(-49\right)}$$

Tuhia te pūtakerua o te 29204.

$$t=\frac{-98±14\sqrt{149}}{2\left(-49\right)}$$

Whakareatia 2 ki te -49.

$$t=\frac{-98±14\sqrt{149}}{-98}$$

Nā, me whakaoti te whārite t=\frac{-98±14\sqrt{149}}{-98} ina he tāpiri te ±. Tāpiri -98 ki te 14\sqrt{149}.

$$t=\frac{14\sqrt{149}-98}{-98}$$

Whakawehe -98+14\sqrt{149} ki te -98.

$$t=-\frac{\sqrt{149}}{7}+1$$

Nā, me whakaoti te whārite t=\frac{-98±14\sqrt{149}}{-98} ina he tango te ±. Tango 14\sqrt{149} mai i -98.

$$t=\frac{-14\sqrt{149}-98}{-98}$$

Whakawehe -98-14\sqrt{149} ki te -98.

$$t=\frac{\sqrt{149}}{7}+1$$

Kua oti te whārite te whakatau.

$$t=-\frac{\sqrt{149}}{7}+1 t=\frac{\sqrt{149}}{7}+1$$

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.

$$-49t^{2}+98t+100=0$$

Me tango 100 mai i ngā taha e rua o te whārite.

$$-49t^{2}+98t+100-100=-100$$

Mā te tango i te 100 i a ia ake anō ka toe ko te 0.

$$-49t^{2}+98t=-100$$

Whakawehea ngā taha e rua ki te -49.

$$\frac{-49t^{2}+98t}{-49}=-\frac{100}{-49}$$

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

$$t^{2}+\frac{98}{-49}t=-\frac{100}{-49}$$

Whakawehe 98 ki te -49.

$$t^{2}-2t=-\frac{100}{-49}$$

Whakawehe -100 ki te -49.

$$t^{2}-2t=\frac{100}{49}$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-2t+1=\frac{100}{49}+1$$

Tāpiri \frac{100}{49} ki te 1.

$$t^{2}-2t+1=\frac{149}{49}$$

Tauwehea t^{2}-2t+1. 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}.

$$\left(t-1\right)^{2}=\frac{149}{49}$$

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

$$\sqrt{\left(t-1\right)^{2}}=\sqrt{\frac{149}{49}}$$

Whakarūnātia.

$$t-1=\frac{\sqrt{149}}{7} t-1=-\frac{\sqrt{149}}{7}$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$t=\frac{\sqrt{149}}{7}+1 t=-\frac{\sqrt{149}}{7}+1$$