sqrt{2x-1}-sqrt{x-1}=sqrt{6-x} eching

$$\sqrt { 2 x - 1 } - \sqrt { x - 1 } = \sqrt { 6 - x }$$

x uchun yechish

$$x=5$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

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

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

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

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

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

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

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

3x ni olish uchun 2x va x ni birlashtirish.

$$3x-1-2\sqrt{2x-1}\sqrt{x-1}-1=\left(\sqrt{6-x}\right)^{2}$$

-2 olish uchun -1 dan 1 ni ayirish.

$$3x-2-2\sqrt{2x-1}\sqrt{x-1}=\left(\sqrt{6-x}\right)^{2}$$

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

$$3x-2-2\sqrt{2x-1}\sqrt{x-1}=6-x$$

Tenglamaning ikkala tarafidan 3x-2 ni ayirish.

$$-2\sqrt{2x-1}\sqrt{x-1}=6-x-\left(3x-2\right)$$

3x-2 teskarisini topish uchun har birining teskarisini toping.

$$-2\sqrt{2x-1}\sqrt{x-1}=6-x-3x+2$$

-4x ni olish uchun -x va -3x ni birlashtirish.

$$-2\sqrt{2x-1}\sqrt{x-1}=6-4x+2$$

8 olish uchun 6 va 2'ni qo'shing.

$$-2\sqrt{2x-1}\sqrt{x-1}=8-4x$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-2\sqrt{2x-1}\sqrt{x-1}\right)^{2}=\left(8-4x\right)^{2}$$

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

$$\left(-2\right)^{2}\left(\sqrt{2x-1}\right)^{2}\left(\sqrt{x-1}\right)^{2}=\left(8-4x\right)^{2}$$

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

$$4\left(\sqrt{2x-1}\right)^{2}\left(\sqrt{x-1}\right)^{2}=\left(8-4x\right)^{2}$$

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

$$4\left(2x-1\right)\left(\sqrt{x-1}\right)^{2}=\left(8-4x\right)^{2}$$

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

$$4\left(2x-1\right)\left(x-1\right)=\left(8-4x\right)^{2}$$

4 ga 2x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(8x-4\right)\left(x-1\right)=\left(8-4x\right)^{2}$$

8x-4 ifodaning har bir elementini x-1 ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$8x^{2}-8x-4x+4=\left(8-4x\right)^{2}$$

-12x ni olish uchun -8x va -4x ni birlashtirish.

$$8x^{2}-12x+4=\left(8-4x\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(8-4x\right)^{2} kengaytirilishi uchun ishlating.

$$8x^{2}-12x+4=64-64x+16x^{2}$$

Ikkala tarafdan 64 ni ayirish.

$$8x^{2}-12x+4-64=-64x+16x^{2}$$

-60 olish uchun 4 dan 64 ni ayirish.

$$8x^{2}-12x-60=-64x+16x^{2}$$

64x ni ikki tarafga qo’shing.

$$8x^{2}-12x-60+64x=16x^{2}$$

52x ni olish uchun -12x va 64x ni birlashtirish.

$$8x^{2}+52x-60=16x^{2}$$

Ikkala tarafdan 16x^{2} ni ayirish.

$$8x^{2}+52x-60-16x^{2}=0$$

-8x^{2} ni olish uchun 8x^{2} va -16x^{2} ni birlashtirish.

$$-8x^{2}+52x-60=0$$

Ikki tarafini 4 ga bo‘ling.

$$-2x^{2}+13x-15=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -2x^{2}+ax+bx-15 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=13 ab=-2\left(-15\right)=30$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 30-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,30 2,15 3,10 5,6$$

Har bir juftlik yigʻindisini hisoblang.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

Yechim – 13 yigʻindisini beruvchi juftlik.

$$a=10 b=3$$

-2x^{2}+13x-15 ni \left(-2x^{2}+10x\right)+\left(3x-15\right) sifatida qaytadan yozish.

$$\left(-2x^{2}+10x\right)+\left(3x-15\right)$$

Birinchi guruhda 2x ni va ikkinchi guruhda -3 ni faktordan chiqaring.

$$2x\left(-x+5\right)-3\left(-x+5\right)$$

Distributiv funktsiyasidan foydalangan holda -x+5 umumiy terminini chiqaring.

$$\left(-x+5\right)\left(2x-3\right)$$

Tenglamani yechish uchun -x+5=0 va 2x-3=0 ni yeching.

$$x=5 x=\frac{3}{2}$$

\sqrt{2x-1}-\sqrt{x-1}=\sqrt{6-x} tenglamasida x uchun 5 ni almashtiring.

$$\sqrt{2\times 5-1}-\sqrt{5-1}=\sqrt{6-5}$$

Qisqartirish. x=5 tenglamani qoniqtiradi.

$$1=1$$

\sqrt{2x-1}-\sqrt{x-1}=\sqrt{6-x} tenglamasida x uchun \frac{3}{2} ni almashtiring.

$$\sqrt{2\times \frac{3}{2}-1}-\sqrt{\frac{3}{2}-1}=\sqrt{6-\frac{3}{2}}$$

Qisqartirish. x=\frac{3}{2} qiymati bu tenglamani qoniqtirmaydi.

$$\frac{1}{2}\times 2^{\frac{1}{2}}=\frac{3}{2}\times 2^{\frac{1}{2}}$$

\sqrt{2x-1}-\sqrt{x-1}=\sqrt{6-x} tenglamasida x uchun 5 ni almashtiring.

$$\sqrt{2\times 5-1}-\sqrt{5-1}=\sqrt{6-5}$$

Qisqartirish. x=5 tenglamani qoniqtiradi.

$$1=1$$

\sqrt{2x-1}-\sqrt{x-1}=\sqrt{6-x} tenglamasi noyob yechimga ega.

$$x=5$$