Whakaoti i te sqrt{3/5}(x+1)+sqrt{5/3}(x-1)=1/15

$$\sqrt { \frac { 3 } { 5 } } ( x + 1 ) + \sqrt { \frac { 5 } { 3 } } ( x - 1 ) = \frac { 1 } { 15 }$$

Whakaoti mō x

$$x=\frac{\sqrt{15}+30}{120}\approx 0.282274861$$

Ngā Upane mō te Whakaoti Whārite Rārangi

Tuhia anō te pūtake rua o te whakawehenga \sqrt{\frac{3}{5}} hei whakawehenga o ngā pūtake rua \frac{\sqrt{3}}{\sqrt{5}}.

$$\frac{\sqrt{3}}{\sqrt{5}}\left(x+1\right)+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

Whakangāwaritia te tauraro o \frac{\sqrt{3}}{\sqrt{5}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{5}.

$$\frac{\sqrt{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\left(x+1\right)+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

Ko te pūrua o \sqrt{5} ko 5.

$$\frac{\sqrt{3}\sqrt{5}}{5}\left(x+1\right)+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

Hei whakarea \sqrt{3} me \sqrt{5}, whakareatia ngā tau i raro i te pūtake rua.

$$\frac{\sqrt{15}}{5}\left(x+1\right)+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

Tuhia te \frac{\sqrt{15}}{5}\left(x+1\right) hei hautanga kotahi.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\sqrt{\frac{5}{3}}\left(x-1\right)=\frac{1}{15}$$

Tuhia anō te pūtake rua o te whakawehenga \sqrt{\frac{5}{3}} hei whakawehenga o ngā pūtake rua \frac{\sqrt{5}}{\sqrt{3}}.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{5}}{\sqrt{3}}\left(x-1\right)=\frac{1}{15}$$

Whakangāwaritia te tauraro o \frac{\sqrt{5}}{\sqrt{3}} mā te whakarea i te taurunga me te tauraro ki te \sqrt{3}.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\left(x-1\right)=\frac{1}{15}$$

Ko te pūrua o \sqrt{3} ko 3.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{5}\sqrt{3}}{3}\left(x-1\right)=\frac{1}{15}$$

Hei whakarea \sqrt{5} me \sqrt{3}, whakareatia ngā tau i raro i te pūtake rua.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{15}}{3}\left(x-1\right)=\frac{1}{15}$$

Tuhia te \frac{\sqrt{15}}{3}\left(x-1\right) hei hautanga kotahi.

$$\frac{\sqrt{15}\left(x+1\right)}{5}+\frac{\sqrt{15}\left(x-1\right)}{3}=\frac{1}{15}$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o 5 me 3 ko 15. Whakareatia \frac{\sqrt{15}\left(x+1\right)}{5} ki te \frac{3}{3}. Whakareatia \frac{\sqrt{15}\left(x-1\right)}{3} ki te \frac{5}{5}.

$$\frac{3\sqrt{15}\left(x+1\right)}{15}+\frac{5\sqrt{15}\left(x-1\right)}{15}=\frac{1}{15}$$

Tā te mea he rite te tauraro o \frac{3\sqrt{15}\left(x+1\right)}{15} me \frac{5\sqrt{15}\left(x-1\right)}{15}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\frac{3\sqrt{15}\left(x+1\right)+5\sqrt{15}\left(x-1\right)}{15}=\frac{1}{15}$$

Mahia ngā whakarea i roto o 3\sqrt{15}\left(x+1\right)+5\sqrt{15}\left(x-1\right).

$$\frac{3\sqrt{15}x+3\sqrt{15}+5\sqrt{15}x-5\sqrt{15}}{15}=\frac{1}{15}$$

Whakakotahitia ngā kupu rite i 3\sqrt{15}x+3\sqrt{15}+5\sqrt{15}x-5\sqrt{15}.

$$\frac{8\sqrt{15}x-2\sqrt{15}}{15}=\frac{1}{15}$$

Me whakarea ngā taha e rua ki te 15.

$$8\sqrt{15}x-2\sqrt{15}=\frac{1}{15}\times 15$$

Me whakakore te 15 me te 15.

$$8\sqrt{15}x-2\sqrt{15}=1$$

Me tāpiri te 2\sqrt{15} ki ngā taha e rua.

$$8\sqrt{15}x=1+2\sqrt{15}$$

He hanga arowhānui tō te whārite.

$$8\sqrt{15}x=2\sqrt{15}+1$$

Whakawehea ngā taha e rua ki te 8\sqrt{15}.

$$\frac{8\sqrt{15}x}{8\sqrt{15}}=\frac{2\sqrt{15}+1}{8\sqrt{15}}$$

Mā te whakawehe ki te 8\sqrt{15} ka wetekia te whakareanga ki te 8\sqrt{15}.

$$x=\frac{2\sqrt{15}+1}{8\sqrt{15}}$$

Whakawehe 1+2\sqrt{15} ki te 8\sqrt{15}.

$$x=\frac{\sqrt{15}}{120}+\frac{1}{4}$$