Réitigh sqrt{3/5}(x+1)+sqrt{5/3}(x-1)=1/15

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

Réitigh do x.

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

Céimeanna chun Cothromóidí Líneacha a Réiteach

Athscríobh fréamh cearnach na roinnte \sqrt{\frac{3}{5}} mar roinnt na bhfréamhacha cearnacha \frac{\sqrt{3}}{\sqrt{5}}.

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

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi \sqrt{5} chun ainmneoir \frac{\sqrt{3}}{\sqrt{5}} a thiontú in uimhir chóimheasta.

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

Is é 5 uimhir chearnach \sqrt{5}.

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

Iolraigh na huimhreacha faoin bhfréamh cearnach chun \sqrt{3} agus \sqrt{5} a iolrú.

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

Scríobh \frac{\sqrt{15}}{5}\left(x+1\right) mar chodán aonair.

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

Athscríobh fréamh cearnach na roinnte \sqrt{\frac{5}{3}} mar roinnt na bhfréamhacha cearnacha \frac{\sqrt{5}}{\sqrt{3}}.

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

Iolraigh an t-uimhreoir agus an t-ainmneoir faoi \sqrt{3} chun ainmneoir \frac{\sqrt{5}}{\sqrt{3}} a thiontú in uimhir chóimheasta.

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

Is é 3 uimhir chearnach \sqrt{3}.

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

Iolraigh na huimhreacha faoin bhfréamh cearnach chun \sqrt{5} agus \sqrt{3} a iolrú.

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

Scríobh \frac{\sqrt{15}}{3}\left(x-1\right) mar chodán aonair.

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

Chun cothromóidí a shuimiú nó a dhealú, fairsingigh iad chun a n-ainmneoirí a mheaitseáil. Is é an t-iolrach is lú coitianta de 5 agus 3 ná 15. Méadaigh \frac{\sqrt{15}\left(x+1\right)}{5} faoi \frac{3}{3}. Méadaigh \frac{\sqrt{15}\left(x-1\right)}{3} faoi \frac{5}{5}.

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

Tá an t-ainmneoir céanna ag \frac{3\sqrt{15}\left(x+1\right)}{15} agus \frac{5\sqrt{15}\left(x-1\right)}{15} agus, mar sin, is féidir iad a shuimiú trína n-uimhreoirí a shuimiú.

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

Déan iolrúcháin in 3\sqrt{15}\left(x+1\right)+5\sqrt{15}\left(x-1\right).

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

Cumaisc téarmaí comhchosúla in: 3\sqrt{15}x+3\sqrt{15}+5\sqrt{15}x-5\sqrt{15}.

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

Iolraigh an dá thaobh faoi 15.

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

Cealaigh 15 agus 15.

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

Cuir 2\sqrt{15} leis an dá thaobh.

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

Tá an chothromóid i bhfoirm chaighdeánach.

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

Roinn an dá thaobh faoi 8\sqrt{15}.

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

Má roinntear é faoi 8\sqrt{15} cuirtear an iolrúchán faoi 8\sqrt{15} ar ceal.

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

Roinn 1+2\sqrt{15} faoi 8\sqrt{15}.

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