frac{2+sqrt{3}}{2-sqrt{3}}=a+bsqrt{3} eching

$$\frac { 2 + \sqrt { 3 } } { 2 - \sqrt { 3 } } = a + b \sqrt { 3 }$$

b uchun yechish

$$b=-\frac{\sqrt{3}\left(a-4\sqrt{3}-7\right)}{3}$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

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

$$\frac{\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}=a+b\sqrt{3}$$

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

$$\frac{\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)}{2^{2}-\left(\sqrt{3}\right)^{2}}=a+b\sqrt{3}$$

2 kvadratini chiqarish. \sqrt{3} kvadratini chiqarish.

$$\frac{\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)}{4-3}=a+b\sqrt{3}$$

1 olish uchun 4 dan 3 ni ayirish.

$$\frac{\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)}{1}=a+b\sqrt{3}$$

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

$$\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)=a+b\sqrt{3}$$

\left(2+\sqrt{3}\right)^{2} hosil qilish uchun 2+\sqrt{3} va 2+\sqrt{3} ni ko'paytirish.

$$\left(2+\sqrt{3}\right)^{2}=a+b\sqrt{3}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2+\sqrt{3}\right)^{2} kengaytirilishi uchun ishlating.

$$4+4\sqrt{3}+\left(\sqrt{3}\right)^{2}=a+b\sqrt{3}$$

\sqrt{3} kvadrati – 3.

$$4+4\sqrt{3}+3=a+b\sqrt{3}$$

7 olish uchun 4 va 3'ni qo'shing.

$$7+4\sqrt{3}=a+b\sqrt{3}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$a+b\sqrt{3}=7+4\sqrt{3}$$

Ikkala tarafdan a ni ayirish.

$$b\sqrt{3}=7+4\sqrt{3}-a$$

Tenglama standart shaklda.

$$\sqrt{3}b=-a+4\sqrt{3}+7$$

Ikki tarafini \sqrt{3} ga bo‘ling.

$$\frac{\sqrt{3}b}{\sqrt{3}}=\frac{-a+4\sqrt{3}+7}{\sqrt{3}}$$

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

$$b=\frac{-a+4\sqrt{3}+7}{\sqrt{3}}$$

4\sqrt{3}-a+7 ni \sqrt{3} ga bo'lish.

$$b=\frac{\sqrt{3}\left(-a+4\sqrt{3}+7\right)}{3}$$

a uchun yechish

$$a=-\sqrt{3}b+4\sqrt{3}+7$$