Whakaoti i te 9t^2-36t+1=0

$$9 { t }^{ 2 } -36t+1 = 0$$

Whakaoti mō t

$$t=\frac{\sqrt{35}}{3}+2\approx 3.972026594$$

$$t=-\frac{\sqrt{35}}{3}+2\approx 0.027973406$$

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.

$$9t^{2}-36t+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -36 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 9}}{2\times 9}$$

Pūrua -36.

$$t=\frac{-\left(-36\right)±\sqrt{1296-4\times 9}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$t=\frac{-\left(-36\right)±\sqrt{1296-36}}{2\times 9}$$

Tāpiri 1296 ki te -36.

$$t=\frac{-\left(-36\right)±\sqrt{1260}}{2\times 9}$$

Tuhia te pūtakerua o te 1260.

$$t=\frac{-\left(-36\right)±6\sqrt{35}}{2\times 9}$$

Ko te tauaro o -36 ko 36.

$$t=\frac{36±6\sqrt{35}}{2\times 9}$$

Whakareatia 2 ki te 9.

$$t=\frac{36±6\sqrt{35}}{18}$$

Nā, me whakaoti te whārite t=\frac{36±6\sqrt{35}}{18} ina he tāpiri te ±. Tāpiri 36 ki te 6\sqrt{35}.

$$t=\frac{6\sqrt{35}+36}{18}$$

Whakawehe 36+6\sqrt{35} ki te 18.

$$t=\frac{\sqrt{35}}{3}+2$$

Nā, me whakaoti te whārite t=\frac{36±6\sqrt{35}}{18} ina he tango te ±. Tango 6\sqrt{35} mai i 36.

$$t=\frac{36-6\sqrt{35}}{18}$$

Whakawehe 36-6\sqrt{35} ki te 18.

$$t=-\frac{\sqrt{35}}{3}+2$$

Kua oti te whārite te whakatau.

$$t=\frac{\sqrt{35}}{3}+2 t=-\frac{\sqrt{35}}{3}+2$$

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.

$$9t^{2}-36t+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$9t^{2}-36t+1-1=-1$$

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$9t^{2}-36t=-1$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9t^{2}-36t}{9}=-\frac{1}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$t^{2}+\left(-\frac{36}{9}\right)t=-\frac{1}{9}$$

Whakawehe -36 ki te 9.

$$t^{2}-4t=-\frac{1}{9}$$

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

$$t^{2}-4t+\left(-2\right)^{2}=-\frac{1}{9}+\left(-2\right)^{2}$$

Pūrua -2.

$$t^{2}-4t+4=-\frac{1}{9}+4$$

Tāpiri -\frac{1}{9} ki te 4.

$$t^{2}-4t+4=\frac{35}{9}$$

Tauwehea t^{2}-4t+4. 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}.

$$\left(t-2\right)^{2}=\frac{35}{9}$$

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

$$\sqrt{\left(t-2\right)^{2}}=\sqrt{\frac{35}{9}}$$

Whakarūnātia.

$$t-2=\frac{\sqrt{35}}{3} t-2=-\frac{\sqrt{35}}{3}$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$t=\frac{\sqrt{35}}{3}+2 t=-\frac{\sqrt{35}}{3}+2$$