Whakaoti i te (5n=n^2-n-1)

$$( 5 n = n ^ { 2 } - n - 1 )$$

Whakaoti mō n

$$n=\sqrt{10}+3\approx 6.16227766$$

$$n=3-\sqrt{10}\approx -0.16227766$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te n^{2} mai i ngā taha e rua.

$$5n-n^{2}=-n-1$$

Me tāpiri te n ki ngā taha e rua.

$$5n-n^{2}+n=-1$$

Pahekotia te 5n me n, ka 6n.

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

Me tāpiri te 1 ki ngā taha e rua.

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

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.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 6 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-6±\sqrt{6^{2}-4\left(-1\right)}}{2\left(-1\right)}$$

Pūrua 6.

$$n=\frac{-6±\sqrt{36-4\left(-1\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$n=\frac{-6±\sqrt{36+4}}{2\left(-1\right)}$$

Tāpiri 36 ki te 4.

$$n=\frac{-6±\sqrt{40}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 40.

$$n=\frac{-6±2\sqrt{10}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$n=\frac{-6±2\sqrt{10}}{-2}$$

Nā, me whakaoti te whārite n=\frac{-6±2\sqrt{10}}{-2} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{10}.

$$n=\frac{2\sqrt{10}-6}{-2}$$

Whakawehe -6+2\sqrt{10} ki te -2.

$$n=3-\sqrt{10}$$

Nā, me whakaoti te whārite n=\frac{-6±2\sqrt{10}}{-2} ina he tango te ±. Tango 2\sqrt{10} mai i -6.

$$n=\frac{-2\sqrt{10}-6}{-2}$$

Whakawehe -6-2\sqrt{10} ki te -2.

$$n=\sqrt{10}+3$$

Kua oti te whārite te whakatau.

$$n=3-\sqrt{10} n=\sqrt{10}+3$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te n^{2} mai i ngā taha e rua.

$$5n-n^{2}=-n-1$$

Me tāpiri te n ki ngā taha e rua.

$$5n-n^{2}+n=-1$$

Pahekotia te 5n me n, ka 6n.

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

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.

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

Whakawehea ngā taha e rua ki te -1.

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

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

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

Whakawehe 6 ki te -1.

$$n^{2}-6n=-\frac{1}{-1}$$

Whakawehe -1 ki te -1.

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

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-6n+\left(-3\right)^{2}=1+\left(-3\right)^{2}$$

Pūrua -3.

$$n^{2}-6n+9=1+9$$

Tāpiri 1 ki te 9.

$$n^{2}-6n+9=10$$

Tauwehea n^{2}-6n+9. 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}.

$$\left(n-3\right)^{2}=10$$

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

$$\sqrt{\left(n-3\right)^{2}}=\sqrt{10}$$

Whakarūnātia.

$$n-3=\sqrt{10} n-3=-\sqrt{10}$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$n=\sqrt{10}+3 n=3-\sqrt{10}$$