Whakaoti i te n^2+2n-1=6

$$n ^ { 2 } + 2 n - 1 = 6$$

Whakaoti mō n

$$n=2\sqrt{2}-1\approx 1.828427125$$

$$n=-2\sqrt{2}-1\approx -3.828427125$$

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.

$$n^{2}+2n-1=6$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$n^{2}+2n-1-6=6-6$$

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

$$n^{2}+2n-1-6=0$$

Tango 6 mai i -1.

$$n^{2}+2n-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 2 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-2±\sqrt{2^{2}-4\left(-7\right)}}{2}$$

Pūrua 2.

$$n=\frac{-2±\sqrt{4-4\left(-7\right)}}{2}$$

Whakareatia -4 ki te -7.

$$n=\frac{-2±\sqrt{4+28}}{2}$$

Tāpiri 4 ki te 28.

$$n=\frac{-2±\sqrt{32}}{2}$$

Tuhia te pūtakerua o te 32.

$$n=\frac{-2±4\sqrt{2}}{2}$$

Nā, me whakaoti te whārite n=\frac{-2±4\sqrt{2}}{2} ina he tāpiri te ±. Tāpiri -2 ki te 4\sqrt{2}.

$$n=\frac{4\sqrt{2}-2}{2}$$

Whakawehe 4\sqrt{2}-2 ki te 2.

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

Nā, me whakaoti te whārite n=\frac{-2±4\sqrt{2}}{2} ina he tango te ±. Tango 4\sqrt{2} mai i -2.

$$n=\frac{-4\sqrt{2}-2}{2}$$

Whakawehe -2-4\sqrt{2} ki te 2.

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

Kua oti te whārite te whakatau.

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

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.

$$n^{2}+2n-1=6$$

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

$$n^{2}+2n-1-\left(-1\right)=6-\left(-1\right)$$

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

$$n^{2}+2n=6-\left(-1\right)$$

Tango -1 mai i 6.

$$n^{2}+2n=7$$

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.

$$n^{2}+2n+1^{2}=7+1^{2}$$

Pūrua 1.

$$n^{2}+2n+1=7+1$$

Tāpiri 7 ki te 1.

$$n^{2}+2n+1=8$$

Tauwehea te n^{2}+2n+1. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+1\right)^{2}=8$$

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

$$\sqrt{\left(n+1\right)^{2}}=\sqrt{8}$$

Whakarūnātia.

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

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

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