sqrt{6x-1}-sqrt{5x+4}=9 eching

$$\sqrt { 6 x - 1 } - \sqrt { 5 x + 4 } = 9$$

x uchun yechish

$$x=18\sqrt{2459}+896\approx 1788,589491312$$

Yechim qadamlari

Tenglamaning ikkala tarafidan -\sqrt{5x+4} ni ayirish.

$$\sqrt{6x-1}=9+\sqrt{5x+4}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{6x-1}\right)^{2}=\left(9+\sqrt{5x+4}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{6x-1} ga hisoblang va 6x-1 ni qiymatni oling.

$$6x-1=\left(9+\sqrt{5x+4}\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(9+\sqrt{5x+4}\right)^{2} kengaytirilishi uchun ishlating.

$$6x-1=81+18\sqrt{5x+4}+\left(\sqrt{5x+4}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{5x+4} ga hisoblang va 5x+4 ni qiymatni oling.

$$6x-1=81+18\sqrt{5x+4}+5x+4$$

85 olish uchun 81 va 4'ni qo'shing.

$$6x-1=85+18\sqrt{5x+4}+5x$$

Tenglamaning ikkala tarafidan 85+5x ni ayirish.

$$6x-1-\left(85+5x\right)=18\sqrt{5x+4}$$

85+5x teskarisini topish uchun har birining teskarisini toping.

$$6x-1-85-5x=18\sqrt{5x+4}$$

-86 olish uchun -1 dan 85 ni ayirish.

$$6x-86-5x=18\sqrt{5x+4}$$

x ni olish uchun 6x va -5x ni birlashtirish.

$$x-86=18\sqrt{5x+4}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(x-86\right)^{2}=\left(18\sqrt{5x+4}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-86\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-172x+7396=\left(18\sqrt{5x+4}\right)^{2}$$

\left(18\sqrt{5x+4}\right)^{2} ni kengaytirish.

$$x^{2}-172x+7396=18^{2}\left(\sqrt{5x+4}\right)^{2}$$

2 daraja ko‘rsatkichini 18 ga hisoblang va 324 ni qiymatni oling.

$$x^{2}-172x+7396=324\left(\sqrt{5x+4}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{5x+4} ga hisoblang va 5x+4 ni qiymatni oling.

$$x^{2}-172x+7396=324\left(5x+4\right)$$

324 ga 5x+4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-172x+7396=1620x+1296$$

Ikkala tarafdan 1620x ni ayirish.

$$x^{2}-172x+7396-1620x=1296$$

-1792x ni olish uchun -172x va -1620x ni birlashtirish.

$$x^{2}-1792x+7396=1296$$

Ikkala tarafdan 1296 ni ayirish.

$$x^{2}-1792x+7396-1296=0$$

6100 olish uchun 7396 dan 1296 ni ayirish.

$$x^{2}-1792x+6100=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -1792 ni b va 6100 ni c bilan almashtiring.

$$x=\frac{-\left(-1792\right)±\sqrt{\left(-1792\right)^{2}-4\times 6100}}{2}$$

-1792 kvadratini chiqarish.

$$x=\frac{-\left(-1792\right)±\sqrt{3211264-4\times 6100}}{2}$$

-4 ni 6100 marotabaga ko'paytirish.

$$x=\frac{-\left(-1792\right)±\sqrt{3211264-24400}}{2}$$

3211264 ni -24400 ga qo'shish.

$$x=\frac{-\left(-1792\right)±\sqrt{3186864}}{2}$$

3186864 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1792\right)±36\sqrt{2459}}{2}$$

-1792 ning teskarisi 1792 ga teng.

$$x=\frac{1792±36\sqrt{2459}}{2}$$

x=\frac{1792±36\sqrt{2459}}{2} tenglamasini yeching, bunda ± musbat. 1792 ni 36\sqrt{2459} ga qo'shish.

$$x=\frac{36\sqrt{2459}+1792}{2}$$

1792+36\sqrt{2459} ni 2 ga bo'lish.

$$x=18\sqrt{2459}+896$$

x=\frac{1792±36\sqrt{2459}}{2} tenglamasini yeching, bunda ± manfiy. 1792 dan 36\sqrt{2459} ni ayirish.

$$x=\frac{1792-36\sqrt{2459}}{2}$$

1792-36\sqrt{2459} ni 2 ga bo'lish.

$$x=896-18\sqrt{2459}$$

Tenglama yechildi.

$$x=18\sqrt{2459}+896 x=896-18\sqrt{2459}$$

\sqrt{6x-1}-\sqrt{5x+4}=9 tenglamasida x uchun 18\sqrt{2459}+896 ni almashtiring.

$$\sqrt{6\left(18\sqrt{2459}+896\right)-1}-\sqrt{5\left(18\sqrt{2459}+896\right)+4}=9$$

Qisqartirish. x=18\sqrt{2459}+896 tenglamani qoniqtiradi.

$$9=9$$

\sqrt{6x-1}-\sqrt{5x+4}=9 tenglamasida x uchun 896-18\sqrt{2459} ni almashtiring.

$$\sqrt{6\left(896-18\sqrt{2459}\right)-1}-\sqrt{5\left(896-18\sqrt{2459}\right)+4}=9$$

Qisqartirish. x=896-18\sqrt{2459} qiymati bu tenglamani qoniqtirmaydi, chunki oʻng va chap tarafdagi belgilar bir-biriga qarama-qarshi.

$$99-2\times 2459^{\frac{1}{2}}=9$$

\sqrt{6x-1}-\sqrt{5x+4}=9 tenglamasida x uchun 18\sqrt{2459}+896 ni almashtiring.

$$\sqrt{6\left(18\sqrt{2459}+896\right)-1}-\sqrt{5\left(18\sqrt{2459}+896\right)+4}=9$$

Qisqartirish. x=18\sqrt{2459}+896 tenglamani qoniqtiradi.

$$9=9$$

\sqrt{6x-1}=\sqrt{5x+4}+9 tenglamasi noyob yechimga ega.

$$x=18\sqrt{2459}+896$$