frac{sqrt{3}-1}{sqrt{3}+1}=a+bsqrt{3} eching

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

b uchun yechish

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

Chiziqli tenglamalarni yechish bo'yicha qadamlar

\frac{\sqrt{3}-1}{\sqrt{3}+1} maxrajini \sqrt{3}-1 orqali surat va maxrajini koʻpaytirish orqali ratsionallashtiring.

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

Hisoblang: \left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

\sqrt{3} kvadratini chiqarish. 1 kvadratini chiqarish.

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

2 olish uchun 3 dan 1 ni ayirish.

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

\left(\sqrt{3}-1\right)^{2} hosil qilish uchun \sqrt{3}-1 va \sqrt{3}-1 ni ko'paytirish.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(\sqrt{3}-1\right)^{2} kengaytirilishi uchun ishlating.

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

\sqrt{3} kvadrati – 3.

$$\frac{3-2\sqrt{3}+1}{2}=a+b\sqrt{3}$$

4 olish uchun 3 va 1'ni qo'shing.

$$\frac{4-2\sqrt{3}}{2}=a+b\sqrt{3}$$

2-\sqrt{3} natijani olish uchun 4-2\sqrt{3} ning har bir ifodasini 2 ga bo‘ling.

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

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

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

Ikkala tarafdan a ni ayirish.

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

Tenglama standart shaklda.

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

Ikki tarafini \sqrt{3} ga bo‘ling.

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

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

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

-\sqrt{3}-a+2 ni \sqrt{3} ga bo'lish.

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

a uchun yechish

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