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

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

x uchun yechish

$$x=\frac{\sqrt{15}+30}{120}\approx 0,282274861$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

\sqrt{\frac{3}{5}} boʻlinmasining kvadrat ildizini \frac{\sqrt{3}}{\sqrt{5}} kvadrat ildizlarining boʻlinmasi sifatida qayta yozing.

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

\frac{\sqrt{3}}{\sqrt{5}} maxrajini \sqrt{5} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

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

\sqrt{5} kvadrati – 5.

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

\sqrt{3} va \sqrt{5} ni koʻpaytirish uchun kvadrat ildiz ichidagi sonlarni koʻpaytiring.

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

\frac{\sqrt{15}}{5}\left(x+1\right) ni yagona kasrga aylantiring.

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

\sqrt{\frac{5}{3}} boʻlinmasining kvadrat ildizini \frac{\sqrt{5}}{\sqrt{3}} kvadrat ildizlarining boʻlinmasi sifatida qayta yozing.

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

\frac{\sqrt{5}}{\sqrt{3}} maxrajini \sqrt{3} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

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

\sqrt{3} kvadrati – 3.

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

\sqrt{5} va \sqrt{3} ni koʻpaytirish uchun kvadrat ildiz ichidagi sonlarni koʻpaytiring.

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

\frac{\sqrt{15}}{3}\left(x-1\right) ni yagona kasrga aylantiring.

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

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 5 va 3 ning eng kichik umumiy karralisi 15. \frac{\sqrt{15}\left(x+1\right)}{5} ni \frac{3}{3} marotabaga ko'paytirish. \frac{\sqrt{15}\left(x-1\right)}{3} ni \frac{5}{5} marotabaga ko'paytirish.

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

\frac{3\sqrt{15}\left(x+1\right)}{15} va \frac{5\sqrt{15}\left(x-1\right)}{15} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

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

3\sqrt{15}\left(x+1\right)+5\sqrt{15}\left(x-1\right) ichidagi ko‘paytirishlarni bajaring.

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

3\sqrt{15}x+3\sqrt{15}+5\sqrt{15}x-5\sqrt{15} kabi iboralarga o‘xshab birlashtiring.

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

Ikkala tarafini 15 ga ko‘paytiring.

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

15 va 15 ni qisqartiring.

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

2\sqrt{15} ni ikki tarafga qo’shing.

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

Tenglama standart shaklda.

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

Ikki tarafini 8\sqrt{15} ga bo‘ling.

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

8\sqrt{15} ga bo'lish 8\sqrt{15} ga ko'paytirishni bekor qiladi.

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

1+2\sqrt{15} ni 8\sqrt{15} ga bo'lish.

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