frac{5+2sqrt{3}}{7+4sqrt{3}}=x+ysqrt{3} eching

$$\frac{ 5+2 \sqrt{ 3 } }{ 7+4 \sqrt{ 3 } } =x+y \sqrt{ 3 }$$

y uchun yechish

$$y=-\frac{\sqrt{3}\left(x+6\sqrt{3}-11\right)}{3}$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

\frac{5+2\sqrt{3}}{7+4\sqrt{3}} maxrajini 7-4\sqrt{3} orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{\left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right)}=x+y\sqrt{3}$$

Hisoblang: \left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{7^{2}-\left(4\sqrt{3}\right)^{2}}=x+y\sqrt{3}$$

2 daraja ko‘rsatkichini 7 ga hisoblang va 49 ni qiymatni oling.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-\left(4\sqrt{3}\right)^{2}}=x+y\sqrt{3}$$

\left(4\sqrt{3}\right)^{2} ni kengaytirish.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-4^{2}\left(\sqrt{3}\right)^{2}}=x+y\sqrt{3}$$

2 daraja ko‘rsatkichini 4 ga hisoblang va 16 ni qiymatni oling.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\left(\sqrt{3}\right)^{2}}=x+y\sqrt{3}$$

\sqrt{3} kvadrati – 3.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\times 3}=x+y\sqrt{3}$$

48 hosil qilish uchun 16 va 3 ni ko'paytirish.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-48}=x+y\sqrt{3}$$

1 olish uchun 49 dan 48 ni ayirish.

$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{1}=x+y\sqrt{3}$$

Har qanday son birga bo‘linganda, natija o‘zi chiqadi.

$$\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)=x+y\sqrt{3}$$

5+2\sqrt{3} ga 7-4\sqrt{3} ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$35-6\sqrt{3}-8\left(\sqrt{3}\right)^{2}=x+y\sqrt{3}$$

\sqrt{3} kvadrati – 3.

$$35-6\sqrt{3}-8\times 3=x+y\sqrt{3}$$

-24 hosil qilish uchun -8 va 3 ni ko'paytirish.

$$35-6\sqrt{3}-24=x+y\sqrt{3}$$

11 olish uchun 35 dan 24 ni ayirish.

$$11-6\sqrt{3}=x+y\sqrt{3}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$x+y\sqrt{3}=11-6\sqrt{3}$$

Ikkala tarafdan x ni ayirish.

$$y\sqrt{3}=11-6\sqrt{3}-x$$

Tenglama standart shaklda.

$$\sqrt{3}y=-x+11-6\sqrt{3}$$

Ikki tarafini \sqrt{3} ga bo‘ling.

$$\frac{\sqrt{3}y}{\sqrt{3}}=\frac{-x+11-6\sqrt{3}}{\sqrt{3}}$$

\sqrt{3} ga bo'lish \sqrt{3} ga ko'paytirishni bekor qiladi.

$$y=\frac{-x+11-6\sqrt{3}}{\sqrt{3}}$$

-6\sqrt{3}-x+11 ni \sqrt{3} ga bo'lish.

$$y=\frac{\sqrt{3}\left(-x+11-6\sqrt{3}\right)}{3}$$

x uchun yechish

$$x=-\sqrt{3}y+11-6\sqrt{3}$$