Whakaoti i te 360/n-1+360/n+2=6

$$\frac { 360 } { n - 1 } + \frac { 360 } { n + 2 } = 6$$

Whakaoti mō n

$$n = \frac{3 \sqrt{1601} + 119}{2} \approx 119.518747071$$

$$n=\frac{119-3\sqrt{1601}}{2}\approx -0.518747071$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe n ki tētahi o ngā uara -2,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(n-1\right)\left(n+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o n-1,n+2.

$$\left(n+2\right)\times 360+\left(n-1\right)\times 360=6\left(n-1\right)\left(n+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te n+2 ki te 360.

$$360n+720+\left(n-1\right)\times 360=6\left(n-1\right)\left(n+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te n-1 ki te 360.

$$360n+720+360n-360=6\left(n-1\right)\left(n+2\right)$$

Pahekotia te 360n me 360n, ka 720n.

$$720n+720-360=6\left(n-1\right)\left(n+2\right)$$

Tangohia te 360 i te 720, ka 360.

$$720n+360=6\left(n-1\right)\left(n+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6 ki te n-1.

$$720n+360=\left(6n-6\right)\left(n+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 6n-6 ki te n+2 ka whakakotahi i ngā kupu rite.

$$720n+360=6n^{2}+6n-12$$

Tangohia te 6n^{2} mai i ngā taha e rua.

$$720n+360-6n^{2}=6n-12$$

Tangohia te 6n mai i ngā taha e rua.

$$720n+360-6n^{2}-6n=-12$$

Pahekotia te 720n me -6n, ka 714n.

$$714n+360-6n^{2}=-12$$

Me tāpiri te 12 ki ngā taha e rua.

$$714n+360-6n^{2}+12=0$$

Tāpirihia te 360 ki te 12, ka 372.

$$714n+372-6n^{2}=0$$

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.

$$-6n^{2}+714n+372=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -6 mō a, 714 mō b, me 372 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-714±\sqrt{714^{2}-4\left(-6\right)\times 372}}{2\left(-6\right)}$$

Pūrua 714.

$$n=\frac{-714±\sqrt{509796-4\left(-6\right)\times 372}}{2\left(-6\right)}$$

Whakareatia -4 ki te -6.

$$n=\frac{-714±\sqrt{509796+24\times 372}}{2\left(-6\right)}$$

Whakareatia 24 ki te 372.

$$n=\frac{-714±\sqrt{509796+8928}}{2\left(-6\right)}$$

Tāpiri 509796 ki te 8928.

$$n=\frac{-714±\sqrt{518724}}{2\left(-6\right)}$$

Tuhia te pūtakerua o te 518724.

$$n=\frac{-714±18\sqrt{1601}}{2\left(-6\right)}$$

Whakareatia 2 ki te -6.

$$n=\frac{-714±18\sqrt{1601}}{-12}$$

Nā, me whakaoti te whārite n=\frac{-714±18\sqrt{1601}}{-12} ina he tāpiri te ±. Tāpiri -714 ki te 18\sqrt{1601}.

$$n=\frac{18\sqrt{1601}-714}{-12}$$

Whakawehe -714+18\sqrt{1601} ki te -12.

$$n=\frac{119-3\sqrt{1601}}{2}$$

Nā, me whakaoti te whārite n=\frac{-714±18\sqrt{1601}}{-12} ina he tango te ±. Tango 18\sqrt{1601} mai i -714.

$$n=\frac{-18\sqrt{1601}-714}{-12}$$

Whakawehe -714-18\sqrt{1601} ki te -12.

$$n=\frac{3\sqrt{1601}+119}{2}$$

Kua oti te whārite te whakatau.

$$n=\frac{119-3\sqrt{1601}}{2} n=\frac{3\sqrt{1601}+119}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe n ki tētahi o ngā uara -2,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(n-1\right)\left(n+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o n-1,n+2.

$$\left(n+2\right)\times 360+\left(n-1\right)\times 360=6\left(n-1\right)\left(n+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te n+2 ki te 360.

$$360n+720+\left(n-1\right)\times 360=6\left(n-1\right)\left(n+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te n-1 ki te 360.

$$360n+720+360n-360=6\left(n-1\right)\left(n+2\right)$$

Pahekotia te 360n me 360n, ka 720n.

$$720n+720-360=6\left(n-1\right)\left(n+2\right)$$

Tangohia te 360 i te 720, ka 360.

$$720n+360=6\left(n-1\right)\left(n+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 6 ki te n-1.

$$720n+360=\left(6n-6\right)\left(n+2\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 6n-6 ki te n+2 ka whakakotahi i ngā kupu rite.

$$720n+360=6n^{2}+6n-12$$

Tangohia te 6n^{2} mai i ngā taha e rua.

$$720n+360-6n^{2}=6n-12$$

Tangohia te 6n mai i ngā taha e rua.

$$720n+360-6n^{2}-6n=-12$$

Pahekotia te 720n me -6n, ka 714n.

$$714n+360-6n^{2}=-12$$

Tangohia te 360 mai i ngā taha e rua.

$$714n-6n^{2}=-12-360$$

Tangohia te 360 i te -12, ka -372.

$$714n-6n^{2}=-372$$

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.

$$-6n^{2}+714n=-372$$

Whakawehea ngā taha e rua ki te -6.

$$\frac{-6n^{2}+714n}{-6}=-\frac{372}{-6}$$

Mā te whakawehe ki te -6 ka wetekia te whakareanga ki te -6.

$$n^{2}+\frac{714}{-6}n=-\frac{372}{-6}$$

Whakawehe 714 ki te -6.

$$n^{2}-119n=-\frac{372}{-6}$$

Whakawehe -372 ki te -6.

$$n^{2}-119n=62$$

Whakawehea te -119, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{119}{2}. Nā, tāpiria te pūrua o te -\frac{119}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-119n+\left(-\frac{119}{2}\right)^{2}=62+\left(-\frac{119}{2}\right)^{2}$$

Pūruatia -\frac{119}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-119n+\frac{14161}{4}=62+\frac{14161}{4}$$

Tāpiri 62 ki te \frac{14161}{4}.

$$n^{2}-119n+\frac{14161}{4}=\frac{14409}{4}$$

Tauwehea te n^{2}-119n+\frac{14161}{4}. 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}.

$$\left(n-\frac{119}{2}\right)^{2}=\frac{14409}{4}$$

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

$$\sqrt{\left(n-\frac{119}{2}\right)^{2}}=\sqrt{\frac{14409}{4}}$$

Whakarūnātia.

$$n-\frac{119}{2}=\frac{3\sqrt{1601}}{2} n-\frac{119}{2}=-\frac{3\sqrt{1601}}{2}$$

Me tāpiri \frac{119}{2} ki ngā taha e rua o te whārite.

$$n=\frac{3\sqrt{1601}+119}{2} n=\frac{119-3\sqrt{1601}}{2}$$