Whakaoti i te (n+3)=(2n)^2+(-1+3)(2n)+(-1)(5)^1

$$( n + 3 ) = ( 2 n ) ^ { 2 } + ( - 1 + 3 ) ( 2 n ) + ( - 1 ) ( 5 ) ^ { 1 }$$

Whakaoti mō n

$$n = \frac{\sqrt{137} - 3}{8} \approx 1.088087489$$

$$n=\frac{-\sqrt{137}-3}{8}\approx -1.838087489$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakarohaina te \left(2n\right)^{2}.

$$n+3=2^{2}n^{2}+\left(-1+3\right)\times 2n-5^{1}$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$n+3=4n^{2}+\left(-1+3\right)\times 2n-5^{1}$$

Tāpirihia te -1 ki te 3, ka 2.

$$n+3=4n^{2}+2\times 2n-5^{1}$$

Whakareatia te 2 ki te 2, ka 4.

$$n+3=4n^{2}+4n-5^{1}$$

Tātaihia te 5 mā te pū o 1, kia riro ko 5.

$$n+3=4n^{2}+4n-5$$

Tangohia te 4n^{2} mai i ngā taha e rua.

$$n+3-4n^{2}=4n-5$$

Tangohia te 4n mai i ngā taha e rua.

$$n+3-4n^{2}-4n=-5$$

Pahekotia te n me -4n, ka -3n.

$$-3n+3-4n^{2}=-5$$

Me tāpiri te 5 ki ngā taha e rua.

$$-3n+3-4n^{2}+5=0$$

Tāpirihia te 3 ki te 5, ka 8.

$$-3n+8-4n^{2}=0$$

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.

$$-4n^{2}-3n+8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4 mō a, -3 mō b, me 8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-4\right)\times 8}}{2\left(-4\right)}$$

Pūrua -3.

$$n=\frac{-\left(-3\right)±\sqrt{9-4\left(-4\right)\times 8}}{2\left(-4\right)}$$

Whakareatia -4 ki te -4.

$$n=\frac{-\left(-3\right)±\sqrt{9+16\times 8}}{2\left(-4\right)}$$

Whakareatia 16 ki te 8.

$$n=\frac{-\left(-3\right)±\sqrt{9+128}}{2\left(-4\right)}$$

Tāpiri 9 ki te 128.

$$n=\frac{-\left(-3\right)±\sqrt{137}}{2\left(-4\right)}$$

Ko te tauaro o -3 ko 3.

$$n=\frac{3±\sqrt{137}}{2\left(-4\right)}$$

Whakareatia 2 ki te -4.

$$n=\frac{3±\sqrt{137}}{-8}$$

Nā, me whakaoti te whārite n=\frac{3±\sqrt{137}}{-8} ina he tāpiri te ±. Tāpiri 3 ki te \sqrt{137}.

$$n=\frac{\sqrt{137}+3}{-8}$$

Whakawehe 3+\sqrt{137} ki te -8.

$$n=\frac{-\sqrt{137}-3}{8}$$

Nā, me whakaoti te whārite n=\frac{3±\sqrt{137}}{-8} ina he tango te ±. Tango \sqrt{137} mai i 3.

$$n=\frac{3-\sqrt{137}}{-8}$$

Whakawehe 3-\sqrt{137} ki te -8.

$$n=\frac{\sqrt{137}-3}{8}$$

Kua oti te whārite te whakatau.

$$n=\frac{-\sqrt{137}-3}{8} n=\frac{\sqrt{137}-3}{8}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakarohaina te \left(2n\right)^{2}.

$$n+3=2^{2}n^{2}+\left(-1+3\right)\times 2n-5^{1}$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$n+3=4n^{2}+\left(-1+3\right)\times 2n-5^{1}$$

Tāpirihia te -1 ki te 3, ka 2.

$$n+3=4n^{2}+2\times 2n-5^{1}$$

Whakareatia te 2 ki te 2, ka 4.

$$n+3=4n^{2}+4n-5^{1}$$

Tātaihia te 5 mā te pū o 1, kia riro ko 5.

$$n+3=4n^{2}+4n-5$$

Tangohia te 4n^{2} mai i ngā taha e rua.

$$n+3-4n^{2}=4n-5$$

Tangohia te 4n mai i ngā taha e rua.

$$n+3-4n^{2}-4n=-5$$

Pahekotia te n me -4n, ka -3n.

$$-3n+3-4n^{2}=-5$$

Tangohia te 3 mai i ngā taha e rua.

$$-3n-4n^{2}=-5-3$$

Tangohia te 3 i te -5, ka -8.

$$-3n-4n^{2}=-8$$

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.

$$-4n^{2}-3n=-8$$

Whakawehea ngā taha e rua ki te -4.

$$\frac{-4n^{2}-3n}{-4}=-\frac{8}{-4}$$

Mā te whakawehe ki te -4 ka wetekia te whakareanga ki te -4.

$$n^{2}+\left(-\frac{3}{-4}\right)n=-\frac{8}{-4}$$

Whakawehe -3 ki te -4.

$$n^{2}+\frac{3}{4}n=-\frac{8}{-4}$$

Whakawehe -8 ki te -4.

$$n^{2}+\frac{3}{4}n=2$$

Whakawehea te \frac{3}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{8}. Nā, tāpiria te pūrua o te \frac{3}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}+\frac{3}{4}n+\left(\frac{3}{8}\right)^{2}=2+\left(\frac{3}{8}\right)^{2}$$

Pūruatia \frac{3}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}+\frac{3}{4}n+\frac{9}{64}=2+\frac{9}{64}$$

Tāpiri 2 ki te \frac{9}{64}.

$$n^{2}+\frac{3}{4}n+\frac{9}{64}=\frac{137}{64}$$

Tauwehea n^{2}+\frac{3}{4}n+\frac{9}{64}. 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}.

$$\left(n+\frac{3}{8}\right)^{2}=\frac{137}{64}$$

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

$$\sqrt{\left(n+\frac{3}{8}\right)^{2}}=\sqrt{\frac{137}{64}}$$

Whakarūnātia.

$$n+\frac{3}{8}=\frac{\sqrt{137}}{8} n+\frac{3}{8}=-\frac{\sqrt{137}}{8}$$

Me tango \frac{3}{8} mai i ngā taha e rua o te whārite.

$$n=\frac{\sqrt{137}-3}{8} n=\frac{-\sqrt{137}-3}{8}$$