frac{2sqrt{2}-3sqrt{7}}{2sqrt{2}+3sqrt{7}}=a+bsqrt{2} eching

$$\frac { 2 \sqrt { 2 } - 3 \sqrt { 7 } } { 2 \sqrt { 2 } + 3 \sqrt { 7 } } = a + b \sqrt { 2 }$$

b uchun yechish

$$b=-\frac{\sqrt{2}\left(55a+71-12\sqrt{14}\right)}{110}$$

Chiziqli tenglamalarni yechish bo'yicha qadamlar

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

$$\frac{\left(2\sqrt{2}-3\sqrt{7}\right)\left(2\sqrt{2}-3\sqrt{7}\right)}{\left(2\sqrt{2}+3\sqrt{7}\right)\left(2\sqrt{2}-3\sqrt{7}\right)}=a+b\sqrt{2}$$

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

$$\frac{\left(2\sqrt{2}-3\sqrt{7}\right)\left(2\sqrt{2}-3\sqrt{7}\right)}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

\left(2\sqrt{2}-3\sqrt{7}\right)^{2} hosil qilish uchun 2\sqrt{2}-3\sqrt{7} va 2\sqrt{2}-3\sqrt{7} ni ko'paytirish.

$$\frac{\left(2\sqrt{2}-3\sqrt{7}\right)^{2}}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(2\sqrt{2}-3\sqrt{7}\right)^{2} kengaytirilishi uchun ishlating.

$$\frac{4\left(\sqrt{2}\right)^{2}-12\sqrt{2}\sqrt{7}+9\left(\sqrt{7}\right)^{2}}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

\sqrt{2} kvadrati – 2.

$$\frac{4\times 2-12\sqrt{2}\sqrt{7}+9\left(\sqrt{7}\right)^{2}}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

8 hosil qilish uchun 4 va 2 ni ko'paytirish.

$$\frac{8-12\sqrt{2}\sqrt{7}+9\left(\sqrt{7}\right)^{2}}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

\sqrt{2} va \sqrt{7} ni koʻpaytirish uchun kvadrat ildiz ichidagi sonlarni koʻpaytiring.

$$\frac{8-12\sqrt{14}+9\left(\sqrt{7}\right)^{2}}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

\sqrt{7} kvadrati – 7.

$$\frac{8-12\sqrt{14}+9\times 7}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

63 hosil qilish uchun 9 va 7 ni ko'paytirish.

$$\frac{8-12\sqrt{14}+63}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

71 olish uchun 8 va 63'ni qo'shing.

$$\frac{71-12\sqrt{14}}{\left(2\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

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

$$\frac{71-12\sqrt{14}}{2^{2}\left(\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

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

$$\frac{71-12\sqrt{14}}{4\left(\sqrt{2}\right)^{2}-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

\sqrt{2} kvadrati – 2.

$$\frac{71-12\sqrt{14}}{4\times 2-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

8 hosil qilish uchun 4 va 2 ni ko'paytirish.

$$\frac{71-12\sqrt{14}}{8-\left(3\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

\left(3\sqrt{7}\right)^{2} ni kengaytirish.

$$\frac{71-12\sqrt{14}}{8-3^{2}\left(\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

2 daraja ko‘rsatkichini 3 ga hisoblang va 9 ni qiymatni oling.

$$\frac{71-12\sqrt{14}}{8-9\left(\sqrt{7}\right)^{2}}=a+b\sqrt{2}$$

\sqrt{7} kvadrati – 7.

$$\frac{71-12\sqrt{14}}{8-9\times 7}=a+b\sqrt{2}$$

63 hosil qilish uchun 9 va 7 ni ko'paytirish.

$$\frac{71-12\sqrt{14}}{8-63}=a+b\sqrt{2}$$

-55 olish uchun 8 dan 63 ni ayirish.

$$\frac{71-12\sqrt{14}}{-55}=a+b\sqrt{2}$$

Surat va maxrajini -1 ga ko‘paytiring.

$$\frac{-71+12\sqrt{14}}{55}=a+b\sqrt{2}$$

-\frac{71}{55}+\frac{12}{55}\sqrt{14} natijani olish uchun -71+12\sqrt{14} ning har bir ifodasini 55 ga bo‘ling.

$$-\frac{71}{55}+\frac{12}{55}\sqrt{14}=a+b\sqrt{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$a+b\sqrt{2}=-\frac{71}{55}+\frac{12}{55}\sqrt{14}$$

Ikkala tarafdan a ni ayirish.

$$b\sqrt{2}=-\frac{71}{55}+\frac{12}{55}\sqrt{14}-a$$

Tenglama standart shaklda.

$$\sqrt{2}b=-a+\frac{12\sqrt{14}}{55}-\frac{71}{55}$$

Ikki tarafini \sqrt{2} ga bo‘ling.

$$\frac{\sqrt{2}b}{\sqrt{2}}=\frac{-a+\frac{12\sqrt{14}}{55}-\frac{71}{55}}{\sqrt{2}}$$

\sqrt{2} ga bo'lish \sqrt{2} ga ko'paytirishni bekor qiladi.

$$b=\frac{-a+\frac{12\sqrt{14}}{55}-\frac{71}{55}}{\sqrt{2}}$$

-\frac{71}{55}+\frac{12\sqrt{14}}{55}-a ni \sqrt{2} ga bo'lish.

$$b=\frac{\sqrt{2}\left(-55a+12\sqrt{14}-71\right)}{110}$$

a uchun yechish

$$a=-\frac{\left(2\sqrt{2}-3\sqrt{7}\right)\left(-3\sqrt{14}b-4b+2\sqrt{2}-3\sqrt{7}\right)}{55}$$