Whakaoti i te 5=10t+5t^2

$$5 = 10 t + 5 t ^ { 2 }$$

Whakaoti mō t (complex solution)

$$t=\sqrt{2}-1\approx 0.414213562$$

$$t=-\left(\sqrt{2}+1\right)\approx -2.414213562$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$10t+5t^{2}=5$$

Tangohia te 5 mai i ngā taha e rua.

$$10t+5t^{2}-5=0$$

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.

$$5t^{2}+10t-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 10 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-10±\sqrt{10^{2}-4\times 5\left(-5\right)}}{2\times 5}$$

Pūrua 10.

$$t=\frac{-10±\sqrt{100-4\times 5\left(-5\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$t=\frac{-10±\sqrt{100-20\left(-5\right)}}{2\times 5}$$

Whakareatia -20 ki te -5.

$$t=\frac{-10±\sqrt{100+100}}{2\times 5}$$

Tāpiri 100 ki te 100.

$$t=\frac{-10±\sqrt{200}}{2\times 5}$$

Tuhia te pūtakerua o te 200.

$$t=\frac{-10±10\sqrt{2}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$t=\frac{-10±10\sqrt{2}}{10}$$

Nā, me whakaoti te whārite t=\frac{-10±10\sqrt{2}}{10} ina he tāpiri te ±. Tāpiri -10 ki te 10\sqrt{2}.

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

Whakawehe -10+10\sqrt{2} ki te 10.

$$t=\sqrt{2}-1$$

Nā, me whakaoti te whārite t=\frac{-10±10\sqrt{2}}{10} ina he tango te ±. Tango 10\sqrt{2} mai i -10.

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

Whakawehe -10-10\sqrt{2} ki te 10.

$$t=-\sqrt{2}-1$$

Kua oti te whārite te whakatau.

$$t=\sqrt{2}-1 t=-\sqrt{2}-1$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$10t+5t^{2}=5$$

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.

$$5t^{2}+10t=5$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5t^{2}+10t}{5}=\frac{5}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$t^{2}+\frac{10}{5}t=\frac{5}{5}$$

Whakawehe 10 ki te 5.

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

Whakawehe 5 ki te 5.

$$t^{2}+2t=1$$

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.

$$t^{2}+2t+1^{2}=1+1^{2}$$

Pūrua 1.

$$t^{2}+2t+1=1+1$$

Tāpiri 1 ki te 1.

$$t^{2}+2t+1=2$$

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}.

$$\left(t+1\right)^{2}=2$$

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

$$\sqrt{\left(t+1\right)^{2}}=\sqrt{2}$$

Whakarūnātia.

$$t+1=\sqrt{2} t+1=-\sqrt{2}$$

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

$$t=\sqrt{2}-1 t=-\sqrt{2}-1$$

Whakaoti mō t

$$t=\sqrt{2}-1\approx 0.414213562$$

$$t=-\sqrt{2}-1\approx -2.414213562$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$10t+5t^{2}=5$$

Tangohia te 5 mai i ngā taha e rua.

$$10t+5t^{2}-5=0$$

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.

$$5t^{2}+10t-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, 10 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-10±\sqrt{10^{2}-4\times 5\left(-5\right)}}{2\times 5}$$

Pūrua 10.

$$t=\frac{-10±\sqrt{100-4\times 5\left(-5\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$t=\frac{-10±\sqrt{100-20\left(-5\right)}}{2\times 5}$$

Whakareatia -20 ki te -5.

$$t=\frac{-10±\sqrt{100+100}}{2\times 5}$$

Tāpiri 100 ki te 100.

$$t=\frac{-10±\sqrt{200}}{2\times 5}$$

Tuhia te pūtakerua o te 200.

$$t=\frac{-10±10\sqrt{2}}{2\times 5}$$

Whakareatia 2 ki te 5.

$$t=\frac{-10±10\sqrt{2}}{10}$$

Nā, me whakaoti te whārite t=\frac{-10±10\sqrt{2}}{10} ina he tāpiri te ±. Tāpiri -10 ki te 10\sqrt{2}.

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

Whakawehe -10+10\sqrt{2} ki te 10.

$$t=\sqrt{2}-1$$

Nā, me whakaoti te whārite t=\frac{-10±10\sqrt{2}}{10} ina he tango te ±. Tango 10\sqrt{2} mai i -10.

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

Whakawehe -10-10\sqrt{2} ki te 10.

$$t=-\sqrt{2}-1$$

Kua oti te whārite te whakatau.

$$t=\sqrt{2}-1 t=-\sqrt{2}-1$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$10t+5t^{2}=5$$

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.

$$5t^{2}+10t=5$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5t^{2}+10t}{5}=\frac{5}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$t^{2}+\frac{10}{5}t=\frac{5}{5}$$

Whakawehe 10 ki te 5.

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

Whakawehe 5 ki te 5.

$$t^{2}+2t=1$$

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.

$$t^{2}+2t+1^{2}=1+1^{2}$$

Pūrua 1.

$$t^{2}+2t+1=1+1$$

Tāpiri 1 ki te 1.

$$t^{2}+2t+1=2$$

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}.

$$\left(t+1\right)^{2}=2$$

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

$$\sqrt{\left(t+1\right)^{2}}=\sqrt{2}$$

Whakarūnātia.

$$t+1=\sqrt{2} t+1=-\sqrt{2}$$

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

$$t=\sqrt{2}-1 t=-\sqrt{2}-1$$