sqrt{x+1}-sqrt{9-x}=sqrt{2x-12} eching

$$\sqrt { x + 1 } - \sqrt { 9 - x } = \sqrt { 2 x - 12 }$$

x uchun yechish

$$x=8$$

$$x=7$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{x+1}-\sqrt{9-x}\right)^{2}=\left(\sqrt{2x-12}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(\sqrt{x+1}-\sqrt{9-x}\right)^{2} kengaytirilishi uchun ishlating.

$$\left(\sqrt{x+1}\right)^{2}-2\sqrt{x+1}\sqrt{9-x}+\left(\sqrt{9-x}\right)^{2}=\left(\sqrt{2x-12}\right)^{2}$$

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

$$x+1-2\sqrt{x+1}\sqrt{9-x}+\left(\sqrt{9-x}\right)^{2}=\left(\sqrt{2x-12}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{9-x} ga hisoblang va 9-x ni qiymatni oling.

$$x+1-2\sqrt{x+1}\sqrt{9-x}+9-x=\left(\sqrt{2x-12}\right)^{2}$$

10 olish uchun 1 va 9'ni qo'shing.

$$x+10-2\sqrt{x+1}\sqrt{9-x}-x=\left(\sqrt{2x-12}\right)^{2}$$

0 ni olish uchun x va -x ni birlashtirish.

$$10-2\sqrt{x+1}\sqrt{9-x}=\left(\sqrt{2x-12}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{2x-12} ga hisoblang va 2x-12 ni qiymatni oling.

$$10-2\sqrt{x+1}\sqrt{9-x}=2x-12$$

Tenglamaning ikkala tarafidan 10 ni ayirish.

$$-2\sqrt{x+1}\sqrt{9-x}=2x-12-10$$

-22 olish uchun -12 dan 10 ni ayirish.

$$-2\sqrt{x+1}\sqrt{9-x}=2x-22$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-2\sqrt{x+1}\sqrt{9-x}\right)^{2}=\left(2x-22\right)^{2}$$

\left(-2\sqrt{x+1}\sqrt{9-x}\right)^{2} ni kengaytirish.

$$\left(-2\right)^{2}\left(\sqrt{x+1}\right)^{2}\left(\sqrt{9-x}\right)^{2}=\left(2x-22\right)^{2}$$

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

$$4\left(\sqrt{x+1}\right)^{2}\left(\sqrt{9-x}\right)^{2}=\left(2x-22\right)^{2}$$

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

$$4\left(x+1\right)\left(\sqrt{9-x}\right)^{2}=\left(2x-22\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{9-x} ga hisoblang va 9-x ni qiymatni oling.

$$4\left(x+1\right)\left(9-x\right)=\left(2x-22\right)^{2}$$

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

$$\left(4x+4\right)\left(9-x\right)=\left(2x-22\right)^{2}$$

4x+4 ifodaning har bir elementini 9-x ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$36x-4x^{2}+36-4x=\left(2x-22\right)^{2}$$

32x ni olish uchun 36x va -4x ni birlashtirish.

$$32x-4x^{2}+36=\left(2x-22\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(2x-22\right)^{2} kengaytirilishi uchun ishlating.

$$32x-4x^{2}+36=4x^{2}-88x+484$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$32x-4x^{2}+36-4x^{2}=-88x+484$$

-8x^{2} ni olish uchun -4x^{2} va -4x^{2} ni birlashtirish.

$$32x-8x^{2}+36=-88x+484$$

88x ni ikki tarafga qo’shing.

$$32x-8x^{2}+36+88x=484$$

120x ni olish uchun 32x va 88x ni birlashtirish.

$$120x-8x^{2}+36=484$$

Ikkala tarafdan 484 ni ayirish.

$$120x-8x^{2}+36-484=0$$

-448 olish uchun 36 dan 484 ni ayirish.

$$120x-8x^{2}-448=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-8x^{2}+120x-448=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -8 ni a, 120 ni b va -448 ni c bilan almashtiring.

$$x=\frac{-120±\sqrt{120^{2}-4\left(-8\right)\left(-448\right)}}{2\left(-8\right)}$$

120 kvadratini chiqarish.

$$x=\frac{-120±\sqrt{14400-4\left(-8\right)\left(-448\right)}}{2\left(-8\right)}$$

-4 ni -8 marotabaga ko'paytirish.

$$x=\frac{-120±\sqrt{14400+32\left(-448\right)}}{2\left(-8\right)}$$

32 ni -448 marotabaga ko'paytirish.

$$x=\frac{-120±\sqrt{14400-14336}}{2\left(-8\right)}$$

14400 ni -14336 ga qo'shish.

$$x=\frac{-120±\sqrt{64}}{2\left(-8\right)}$$

64 ning kvadrat ildizini chiqarish.

$$x=\frac{-120±8}{2\left(-8\right)}$$

2 ni -8 marotabaga ko'paytirish.

$$x=\frac{-120±8}{-16}$$

x=\frac{-120±8}{-16} tenglamasini yeching, bunda ± musbat. -120 ni 8 ga qo'shish.

$$x=-\frac{112}{-16}$$

-112 ni -16 ga bo'lish.

$$x=7$$

x=\frac{-120±8}{-16} tenglamasini yeching, bunda ± manfiy. -120 dan 8 ni ayirish.

$$x=-\frac{128}{-16}$$

-128 ni -16 ga bo'lish.

$$x=8$$

Tenglama yechildi.

$$x=7 x=8$$

\sqrt{x+1}-\sqrt{9-x}=\sqrt{2x-12} tenglamasida x uchun 7 ni almashtiring.

$$\sqrt{7+1}-\sqrt{9-7}=\sqrt{2\times 7-12}$$

Qisqartirish. x=7 tenglamani qoniqtiradi.

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

\sqrt{x+1}-\sqrt{9-x}=\sqrt{2x-12} tenglamasida x uchun 8 ni almashtiring.

$$\sqrt{8+1}-\sqrt{9-8}=\sqrt{2\times 8-12}$$

Qisqartirish. x=8 tenglamani qoniqtiradi.

$$2=2$$

\sqrt{x+1}-\sqrt{9-x}=\sqrt{2x-12} boʻyicha barcha yechimlar roʻyxati.

$$x=7 x=8$$