Whakaoti i te n/3+n=sqrt{3/8}*3

$$\frac{ n }{ 3+n } = \sqrt{ \frac{ 3 }{ 8 } } \times 3$$

Whakaoti mō n

$$n=\frac{-18\sqrt{6}-81}{19}\approx -6.583727125$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tē taea kia ōrite te tāupe n ki -3 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te n+3.

$$n=3\sqrt{\frac{3}{8}}\left(n+3\right)$$

Tuhia anō te pūtake rua o te whakawehenga \sqrt{\frac{3}{8}} hei whakawehenga o ngā pūtake rua \frac{\sqrt{3}}{\sqrt{8}}.

$$n=3\times \frac{\sqrt{3}}{\sqrt{8}}\left(n+3\right)$$

Tauwehea te 8=2^{2}\times 2. Tuhia anō te pūtake rua o te hua \sqrt{2^{2}\times 2} hei hua o ngā pūtake rua \sqrt{2^{2}}\sqrt{2}. Tuhia te pūtakerua o te 2^{2}.

$$n=3\times \frac{\sqrt{3}}{2\sqrt{2}}\left(n+3\right)$$

Whakangāwaritia te tauraro o \frac{\sqrt{3}}{2\sqrt{2}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{2}.

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

Ko te pūrua o \sqrt{2} ko 2.

$$n=3\times \frac{\sqrt{3}\sqrt{2}}{2\times 2}\left(n+3\right)$$

Hei whakarea \sqrt{3} me \sqrt{2}, whakareatia ngā tau i raro i te pūtake rua.

$$n=3\times \frac{\sqrt{6}}{2\times 2}\left(n+3\right)$$

Whakareatia te 2 ki te 2, ka 4.

$$n=3\times \frac{\sqrt{6}}{4}\left(n+3\right)$$

Tuhia te 3\times \frac{\sqrt{6}}{4} hei hautanga kotahi.

$$n=\frac{3\sqrt{6}}{4}\left(n+3\right)$$

Tuhia te \frac{3\sqrt{6}}{4}\left(n+3\right) hei hautanga kotahi.

$$n=\frac{3\sqrt{6}\left(n+3\right)}{4}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3\sqrt{6} ki te n+3.

$$n=\frac{3\sqrt{6}n+9\sqrt{6}}{4}$$

Tangohia te \frac{3\sqrt{6}n+9\sqrt{6}}{4} mai i ngā taha e rua.

$$n-\frac{3\sqrt{6}n+9\sqrt{6}}{4}=0$$

Whakareatia ngā taha e rua o te whārite ki te 4.

$$4n-\left(3\sqrt{6}n+9\sqrt{6}\right)=0$$

Hei kimi i te tauaro o 3\sqrt{6}n+9\sqrt{6}, kimihia te tauaro o ia taurangi.

$$4n-3\sqrt{6}n-9\sqrt{6}=0$$

Me tāpiri te 9\sqrt{6} ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$4n-3\sqrt{6}n=9\sqrt{6}$$

Pahekotia ngā kīanga tau katoa e whai ana i te n.

$$\left(4-3\sqrt{6}\right)n=9\sqrt{6}$$

Whakawehea ngā taha e rua ki te 4-3\sqrt{6}.

$$\frac{\left(4-3\sqrt{6}\right)n}{4-3\sqrt{6}}=\frac{9\sqrt{6}}{4-3\sqrt{6}}$$

Mā te whakawehe ki te 4-3\sqrt{6} ka wetekia te whakareanga ki te 4-3\sqrt{6}.

$$n=\frac{9\sqrt{6}}{4-3\sqrt{6}}$$

Whakawehe 9\sqrt{6} ki te 4-3\sqrt{6}.

$$n=\frac{-18\sqrt{6}-81}{19}$$