sqrt{5x-1}-sqrt{3x-2}=sqrt{x-1} eching

$$\sqrt { 5 x - 1 } - \sqrt { 3 x - 2 } = \sqrt { x - 1 }$$

x uchun yechish

$$x=2$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{5x-1}-\sqrt{3x-2}\right)^{2}=\left(\sqrt{x-1}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(\sqrt{5x-1}-\sqrt{3x-2}\right)^{2} kengaytirilishi uchun ishlating.

$$\left(\sqrt{5x-1}\right)^{2}-2\sqrt{5x-1}\sqrt{3x-2}+\left(\sqrt{3x-2}\right)^{2}=\left(\sqrt{x-1}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{5x-1} ga hisoblang va 5x-1 ni qiymatni oling.

$$5x-1-2\sqrt{5x-1}\sqrt{3x-2}+\left(\sqrt{3x-2}\right)^{2}=\left(\sqrt{x-1}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{3x-2} ga hisoblang va 3x-2 ni qiymatni oling.

$$5x-1-2\sqrt{5x-1}\sqrt{3x-2}+3x-2=\left(\sqrt{x-1}\right)^{2}$$

8x ni olish uchun 5x va 3x ni birlashtirish.

$$8x-1-2\sqrt{5x-1}\sqrt{3x-2}-2=\left(\sqrt{x-1}\right)^{2}$$

-3 olish uchun -1 dan 2 ni ayirish.

$$8x-3-2\sqrt{5x-1}\sqrt{3x-2}=\left(\sqrt{x-1}\right)^{2}$$

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

$$8x-3-2\sqrt{5x-1}\sqrt{3x-2}=x-1$$

Tenglamaning ikkala tarafidan 8x-3 ni ayirish.

$$-2\sqrt{5x-1}\sqrt{3x-2}=x-1-\left(8x-3\right)$$

8x-3 teskarisini topish uchun har birining teskarisini toping.

$$-2\sqrt{5x-1}\sqrt{3x-2}=x-1-8x+3$$

-7x ni olish uchun x va -8x ni birlashtirish.

$$-2\sqrt{5x-1}\sqrt{3x-2}=-7x-1+3$$

2 olish uchun -1 va 3'ni qo'shing.

$$-2\sqrt{5x-1}\sqrt{3x-2}=-7x+2$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-2\sqrt{5x-1}\sqrt{3x-2}\right)^{2}=\left(-7x+2\right)^{2}$$

\left(-2\sqrt{5x-1}\sqrt{3x-2}\right)^{2} ni kengaytirish.

$$\left(-2\right)^{2}\left(\sqrt{5x-1}\right)^{2}\left(\sqrt{3x-2}\right)^{2}=\left(-7x+2\right)^{2}$$

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

$$4\left(\sqrt{5x-1}\right)^{2}\left(\sqrt{3x-2}\right)^{2}=\left(-7x+2\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{5x-1} ga hisoblang va 5x-1 ni qiymatni oling.

$$4\left(5x-1\right)\left(\sqrt{3x-2}\right)^{2}=\left(-7x+2\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{3x-2} ga hisoblang va 3x-2 ni qiymatni oling.

$$4\left(5x-1\right)\left(3x-2\right)=\left(-7x+2\right)^{2}$$

4 ga 5x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(20x-4\right)\left(3x-2\right)=\left(-7x+2\right)^{2}$$

20x-4 ifodaning har bir elementini 3x-2 ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$60x^{2}-40x-12x+8=\left(-7x+2\right)^{2}$$

-52x ni olish uchun -40x va -12x ni birlashtirish.

$$60x^{2}-52x+8=\left(-7x+2\right)^{2}$$

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

$$60x^{2}-52x+8=49x^{2}-28x+4$$

Ikkala tarafdan 49x^{2} ni ayirish.

$$60x^{2}-52x+8-49x^{2}=-28x+4$$

11x^{2} ni olish uchun 60x^{2} va -49x^{2} ni birlashtirish.

$$11x^{2}-52x+8=-28x+4$$

28x ni ikki tarafga qo’shing.

$$11x^{2}-52x+8+28x=4$$

-24x ni olish uchun -52x va 28x ni birlashtirish.

$$11x^{2}-24x+8=4$$

Ikkala tarafdan 4 ni ayirish.

$$11x^{2}-24x+8-4=0$$

4 olish uchun 8 dan 4 ni ayirish.

$$11x^{2}-24x+4=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 11x^{2}+ax+bx+4 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-24 ab=11\times 4=44$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 44-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-44 -2,-22 -4,-11$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-44=-45 -2-22=-24 -4-11=-15$$

Yechim – -24 yigʻindisini beruvchi juftlik.

$$a=-22 b=-2$$

11x^{2}-24x+4 ni \left(11x^{2}-22x\right)+\left(-2x+4\right) sifatida qaytadan yozish.

$$\left(11x^{2}-22x\right)+\left(-2x+4\right)$$

Birinchi guruhda 11x ni va ikkinchi guruhda -2 ni faktordan chiqaring.

$$11x\left(x-2\right)-2\left(x-2\right)$$

Distributiv funktsiyasidan foydalangan holda x-2 umumiy terminini chiqaring.

$$\left(x-2\right)\left(11x-2\right)$$

Tenglamani yechish uchun x-2=0 va 11x-2=0 ni yeching.

$$x=2 x=\frac{2}{11}$$

\sqrt{5x-1}-\sqrt{3x-2}=\sqrt{x-1} tenglamasida x uchun \frac{2}{11} ni almashtiring. \sqrt{5\times \frac{2}{11}-1} ifodasi noaniq, chunki ildiz ostidagi qiymat manfiy boʻlishi mumkin emas.

$$\sqrt{5\times \frac{2}{11}-1}-\sqrt{3\times \frac{2}{11}-2}=\sqrt{\frac{2}{11}-1}$$

\sqrt{5x-1}-\sqrt{3x-2}=\sqrt{x-1} tenglamasida x uchun 2 ni almashtiring.

$$\sqrt{5\times 2-1}-\sqrt{3\times 2-2}=\sqrt{2-1}$$

Qisqartirish. x=2 tenglamani qoniqtiradi.

$$1=1$$

\sqrt{5x-1}-\sqrt{3x-2}=\sqrt{x-1} tenglamasi noyob yechimga ega.

$$x=2$$