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

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

b uchun yechish

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

Chiziqli tenglamalarni yechish bo'yicha qadamlar

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

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

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

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

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

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

-1 olish uchun 3 dan 4 ni ayirish.

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

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

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

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

$$\frac{\left(\sqrt{3}\right)^{2}+4\sqrt{3}+4}{-1}=a+b\sqrt{3}$$

\sqrt{3} kvadrati – 3.

$$\frac{3+4\sqrt{3}+4}{-1}=a+b\sqrt{3}$$

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

$$\frac{7+4\sqrt{3}}{-1}=a+b\sqrt{3}$$

Istalgan sonni -1 ga boʻlsangiz, uning qarama-qarshisi chiqadi. 7+4\sqrt{3} teskarisini topish uchun har birining teskarisini toping.

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

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

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

a uchun yechish

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