{left(10/3right)}^2+{left(frac{2sqrt{73}}{3}right)}^2=2{left(frac{sqrt{52}}{3}right)}^2+2x^2 eching

$${ \left( \frac{ 10 }{ 3 } \right) }^{ 2 } + { \left( \frac{ 2 \sqrt{ 73 } }{ 3 } \right) }^{ 2 } =2 { \left( \frac{ \sqrt{ 52 } }{ 3 } \right) }^{ 2 } +2 { x }^{ 2 }$$

x uchun yechish

$$x=4$$

$$x=-4$$

Kvadratlar farqidan foydalangan qadamlar

2 daraja ko‘rsatkichini \frac{10}{3} ga hisoblang va \frac{100}{9} ni qiymatni oling.

$$\frac{100}{9}+\left(\frac{2\sqrt{73}}{3}\right)^{2}=2\times \left(\frac{\sqrt{52}}{3}\right)^{2}+2x^{2}$$

\frac{2\sqrt{73}}{3}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{100}{9}+\frac{\left(2\sqrt{73}\right)^{2}}{3^{2}}=2\times \left(\frac{\sqrt{52}}{3}\right)^{2}+2x^{2}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 3^{2} ni kengaytirish.

$$\frac{100}{9}+\frac{\left(2\sqrt{73}\right)^{2}}{9}=2\times \left(\frac{\sqrt{52}}{3}\right)^{2}+2x^{2}$$

\frac{100}{9} va \frac{\left(2\sqrt{73}\right)^{2}}{9} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=2\times \left(\frac{\sqrt{52}}{3}\right)^{2}+2x^{2}$$

Faktor: 52=2^{2}\times 13. \sqrt{2^{2}\times 13} koʻpaytmasining kvadrat ildizini \sqrt{2^{2}}\sqrt{13} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 2^{2} ning kvadrat ildizini chiqarish.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=2\times \left(\frac{2\sqrt{13}}{3}\right)^{2}+2x^{2}$$

\frac{2\sqrt{13}}{3}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=2\times \frac{\left(2\sqrt{13}\right)^{2}}{3^{2}}+2x^{2}$$

2\times \frac{\left(2\sqrt{13}\right)^{2}}{3^{2}} ni yagona kasrga aylantiring.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}}{3^{2}}+2x^{2}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 2x^{2} ni \frac{3^{2}}{3^{2}} marotabaga ko'paytirish.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}}{3^{2}}+\frac{2x^{2}\times 3^{2}}{3^{2}}$$

\frac{2\times \left(2\sqrt{13}\right)^{2}}{3^{2}} va \frac{2x^{2}\times 3^{2}}{3^{2}} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

\left(2\sqrt{73}\right)^{2} ni kengaytirish.

$$\frac{100+2^{2}\left(\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

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

$$\frac{100+4\left(\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

\sqrt{73} kvadrati – 73.

$$\frac{100+4\times 73}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

292 hosil qilish uchun 4 va 73 ni ko'paytirish.

$$\frac{100+292}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

392 olish uchun 100 va 292'ni qo'shing.

$$\frac{392}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

\left(2\sqrt{13}\right)^{2} ni kengaytirish.

$$\frac{392}{9}=\frac{2\times 2^{2}\left(\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

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

$$\frac{392}{9}=\frac{2\times 4\left(\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

\sqrt{13} kvadrati – 13.

$$\frac{392}{9}=\frac{2\times 4\times 13+2x^{2}\times 3^{2}}{3^{2}}$$

52 hosil qilish uchun 4 va 13 ni ko'paytirish.

$$\frac{392}{9}=\frac{2\times 52+2x^{2}\times 3^{2}}{3^{2}}$$

104 hosil qilish uchun 2 va 52 ni ko'paytirish.

$$\frac{392}{9}=\frac{104+2x^{2}\times 3^{2}}{3^{2}}$$

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

$$\frac{392}{9}=\frac{104+2x^{2}\times 9}{3^{2}}$$

18 hosil qilish uchun 2 va 9 ni ko'paytirish.

$$\frac{392}{9}=\frac{104+18x^{2}}{3^{2}}$$

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

$$\frac{392}{9}=\frac{104+18x^{2}}{9}$$

\frac{104}{9}+2x^{2} natijani olish uchun 104+18x^{2} ning har bir ifodasini 9 ga bo‘ling.

$$\frac{392}{9}=\frac{104}{9}+2x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$\frac{104}{9}+2x^{2}=\frac{392}{9}$$

Ikkala tarafdan \frac{392}{9} ni ayirish.

$$\frac{104}{9}+2x^{2}-\frac{392}{9}=0$$

-32 olish uchun \frac{104}{9} dan \frac{392}{9} ni ayirish.

$$-32+2x^{2}=0$$

Ikki tarafini 2 ga bo‘ling.

$$-16+x^{2}=0$$

Hisoblang: -16+x^{2}. -16+x^{2} ni x^{2}-4^{2} sifatida qaytadan yozish. Kvadratlarning farqini ushbu formula bilan hisoblash mumkin: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(x-4\right)\left(x+4\right)=0$$

Tenglamani yechish uchun x-4=0 va x+4=0 ni yeching.

$$x=4 x=-4$$

Kvadrat ildizini topish bo'yicha qadamlar

2 daraja ko‘rsatkichini \frac{10}{3} ga hisoblang va \frac{100}{9} ni qiymatni oling.

$$\frac{100}{9}+\left(\frac{2\sqrt{73}}{3}\right)^{2}=2\times \left(\frac{\sqrt{52}}{3}\right)^{2}+2x^{2}$$

\frac{2\sqrt{73}}{3}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{100}{9}+\frac{\left(2\sqrt{73}\right)^{2}}{3^{2}}=2\times \left(\frac{\sqrt{52}}{3}\right)^{2}+2x^{2}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 3^{2} ni kengaytirish.

$$\frac{100}{9}+\frac{\left(2\sqrt{73}\right)^{2}}{9}=2\times \left(\frac{\sqrt{52}}{3}\right)^{2}+2x^{2}$$

\frac{100}{9} va \frac{\left(2\sqrt{73}\right)^{2}}{9} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=2\times \left(\frac{\sqrt{52}}{3}\right)^{2}+2x^{2}$$

Faktor: 52=2^{2}\times 13. \sqrt{2^{2}\times 13} koʻpaytmasining kvadrat ildizini \sqrt{2^{2}}\sqrt{13} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 2^{2} ning kvadrat ildizini chiqarish.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=2\times \left(\frac{2\sqrt{13}}{3}\right)^{2}+2x^{2}$$

\frac{2\sqrt{13}}{3}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=2\times \frac{\left(2\sqrt{13}\right)^{2}}{3^{2}}+2x^{2}$$

2\times \frac{\left(2\sqrt{13}\right)^{2}}{3^{2}} ni yagona kasrga aylantiring.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}}{3^{2}}+2x^{2}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 2x^{2} ni \frac{3^{2}}{3^{2}} marotabaga ko'paytirish.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}}{3^{2}}+\frac{2x^{2}\times 3^{2}}{3^{2}}$$

\frac{2\times \left(2\sqrt{13}\right)^{2}}{3^{2}} va \frac{2x^{2}\times 3^{2}}{3^{2}} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

\left(2\sqrt{73}\right)^{2} ni kengaytirish.

$$\frac{100+2^{2}\left(\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

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

$$\frac{100+4\left(\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

\sqrt{73} kvadrati – 73.

$$\frac{100+4\times 73}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

292 hosil qilish uchun 4 va 73 ni ko'paytirish.

$$\frac{100+292}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

392 olish uchun 100 va 292'ni qo'shing.

$$\frac{392}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

\left(2\sqrt{13}\right)^{2} ni kengaytirish.

$$\frac{392}{9}=\frac{2\times 2^{2}\left(\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

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

$$\frac{392}{9}=\frac{2\times 4\left(\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

\sqrt{13} kvadrati – 13.

$$\frac{392}{9}=\frac{2\times 4\times 13+2x^{2}\times 3^{2}}{3^{2}}$$

52 hosil qilish uchun 4 va 13 ni ko'paytirish.

$$\frac{392}{9}=\frac{2\times 52+2x^{2}\times 3^{2}}{3^{2}}$$

104 hosil qilish uchun 2 va 52 ni ko'paytirish.

$$\frac{392}{9}=\frac{104+2x^{2}\times 3^{2}}{3^{2}}$$

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

$$\frac{392}{9}=\frac{104+2x^{2}\times 9}{3^{2}}$$

18 hosil qilish uchun 2 va 9 ni ko'paytirish.

$$\frac{392}{9}=\frac{104+18x^{2}}{3^{2}}$$

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

$$\frac{392}{9}=\frac{104+18x^{2}}{9}$$

\frac{104}{9}+2x^{2} natijani olish uchun 104+18x^{2} ning har bir ifodasini 9 ga bo‘ling.

$$\frac{392}{9}=\frac{104}{9}+2x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$\frac{104}{9}+2x^{2}=\frac{392}{9}$$

Ikkala tarafdan \frac{104}{9} ni ayirish.

$$2x^{2}=\frac{392}{9}-\frac{104}{9}$$

32 olish uchun \frac{392}{9} dan \frac{104}{9} ni ayirish.

$$2x^{2}=32$$

Ikki tarafini 2 ga bo‘ling.

$$x^{2}=\frac{32}{2}$$

16 ni olish uchun 32 ni 2 ga bo‘ling.

$$x^{2}=16$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$x=4 x=-4$$

Kvadrat tenglamalaridan foydalanish qoidalari

2 daraja ko‘rsatkichini \frac{10}{3} ga hisoblang va \frac{100}{9} ni qiymatni oling.

$$\frac{100}{9}+\left(\frac{2\sqrt{73}}{3}\right)^{2}=2\times \left(\frac{\sqrt{52}}{3}\right)^{2}+2x^{2}$$

\frac{2\sqrt{73}}{3}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{100}{9}+\frac{\left(2\sqrt{73}\right)^{2}}{3^{2}}=2\times \left(\frac{\sqrt{52}}{3}\right)^{2}+2x^{2}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 3^{2} ni kengaytirish.

$$\frac{100}{9}+\frac{\left(2\sqrt{73}\right)^{2}}{9}=2\times \left(\frac{\sqrt{52}}{3}\right)^{2}+2x^{2}$$

\frac{100}{9} va \frac{\left(2\sqrt{73}\right)^{2}}{9} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=2\times \left(\frac{\sqrt{52}}{3}\right)^{2}+2x^{2}$$

Faktor: 52=2^{2}\times 13. \sqrt{2^{2}\times 13} koʻpaytmasining kvadrat ildizini \sqrt{2^{2}}\sqrt{13} kvadrat ildizlarining koʻpaytmasi sifatida qayta yozing. 2^{2} ning kvadrat ildizini chiqarish.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=2\times \left(\frac{2\sqrt{13}}{3}\right)^{2}+2x^{2}$$

\frac{2\sqrt{13}}{3}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=2\times \frac{\left(2\sqrt{13}\right)^{2}}{3^{2}}+2x^{2}$$

2\times \frac{\left(2\sqrt{13}\right)^{2}}{3^{2}} ni yagona kasrga aylantiring.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}}{3^{2}}+2x^{2}$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 2x^{2} ni \frac{3^{2}}{3^{2}} marotabaga ko'paytirish.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}}{3^{2}}+\frac{2x^{2}\times 3^{2}}{3^{2}}$$

\frac{2\times \left(2\sqrt{13}\right)^{2}}{3^{2}} va \frac{2x^{2}\times 3^{2}}{3^{2}} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{100+\left(2\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

\left(2\sqrt{73}\right)^{2} ni kengaytirish.

$$\frac{100+2^{2}\left(\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

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

$$\frac{100+4\left(\sqrt{73}\right)^{2}}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

\sqrt{73} kvadrati – 73.

$$\frac{100+4\times 73}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

292 hosil qilish uchun 4 va 73 ni ko'paytirish.

$$\frac{100+292}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

392 olish uchun 100 va 292'ni qo'shing.

$$\frac{392}{9}=\frac{2\times \left(2\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

\left(2\sqrt{13}\right)^{2} ni kengaytirish.

$$\frac{392}{9}=\frac{2\times 2^{2}\left(\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

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

$$\frac{392}{9}=\frac{2\times 4\left(\sqrt{13}\right)^{2}+2x^{2}\times 3^{2}}{3^{2}}$$

\sqrt{13} kvadrati – 13.

$$\frac{392}{9}=\frac{2\times 4\times 13+2x^{2}\times 3^{2}}{3^{2}}$$

52 hosil qilish uchun 4 va 13 ni ko'paytirish.

$$\frac{392}{9}=\frac{2\times 52+2x^{2}\times 3^{2}}{3^{2}}$$

104 hosil qilish uchun 2 va 52 ni ko'paytirish.

$$\frac{392}{9}=\frac{104+2x^{2}\times 3^{2}}{3^{2}}$$

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

$$\frac{392}{9}=\frac{104+2x^{2}\times 9}{3^{2}}$$

18 hosil qilish uchun 2 va 9 ni ko'paytirish.

$$\frac{392}{9}=\frac{104+18x^{2}}{3^{2}}$$

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

$$\frac{392}{9}=\frac{104+18x^{2}}{9}$$

\frac{104}{9}+2x^{2} natijani olish uchun 104+18x^{2} ning har bir ifodasini 9 ga bo‘ling.

$$\frac{392}{9}=\frac{104}{9}+2x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$\frac{104}{9}+2x^{2}=\frac{392}{9}$$

Ikkala tarafdan \frac{392}{9} ni ayirish.

$$\frac{104}{9}+2x^{2}-\frac{392}{9}=0$$

-32 olish uchun \frac{104}{9} dan \frac{392}{9} ni ayirish.

$$-32+2x^{2}=0$$

Bu kabi kvadrat tenglamalarni x^{2} sharti bilan, biroq x shartisiz hamon kvadrat tenglamasidan foydalanib yechish mumkin, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, ular standart formulaga solingandan so'ng: ax^{2}+bx+c=0.

$$2x^{2}-32=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 0 ni b va -32 ni c bilan almashtiring.

$$x=\frac{0±\sqrt{0^{2}-4\times 2\left(-32\right)}}{2\times 2}$$

0 kvadratini chiqarish.

$$x=\frac{0±\sqrt{-4\times 2\left(-32\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{0±\sqrt{-8\left(-32\right)}}{2\times 2}$$

-8 ni -32 marotabaga ko'paytirish.

$$x=\frac{0±\sqrt{256}}{2\times 2}$$

256 ning kvadrat ildizini chiqarish.

$$x=\frac{0±16}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{0±16}{4}$$

x=\frac{0±16}{4} tenglamasini yeching, bunda ± musbat. 16 ni 4 ga bo'lish.

$$x=4$$

x=\frac{0±16}{4} tenglamasini yeching, bunda ± manfiy. -16 ni 4 ga bo'lish.

$$x=-4$$

Tenglama yechildi.

$$x=4 x=-4$$