2sqrt{9x}-6=10-2sqrt{x} eching

$$2 \sqrt { 9 x } - 6 = 10 - 2 \sqrt { x }$$

x uchun yechish

$$x=4$$

Yechim qadamlari

Tenglamaning ikkala tarafidan -6 ni ayirish.

$$2\sqrt{9x}=10-2\sqrt{x}+6$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(2\sqrt{9x}\right)^{2}=\left(10-2\sqrt{x}+6\right)^{2}$$

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

$$2^{2}\left(\sqrt{9x}\right)^{2}=\left(10-2\sqrt{x}+6\right)^{2}$$

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

$$4\left(\sqrt{9x}\right)^{2}=\left(10-2\sqrt{x}+6\right)^{2}$$

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

$$4\times 9x=\left(10-2\sqrt{x}+6\right)^{2}$$

36 hosil qilish uchun 4 va 9 ni ko'paytirish.

$$36x=\left(10-2\sqrt{x}+6\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(10-2\sqrt{x}+6\right)^{2} kengaytirilishi uchun ishlating.

$$36x=\left(10-2\sqrt{x}\right)^{2}+12\left(10-2\sqrt{x}\right)+36$$

Ikkala tarafdan \left(10-2\sqrt{x}\right)^{2} ni ayirish.

$$36x-\left(10-2\sqrt{x}\right)^{2}=12\left(10-2\sqrt{x}\right)+36$$

Ikkala tarafdan 12\left(10-2\sqrt{x}\right) ni ayirish.

$$36x-\left(10-2\sqrt{x}\right)^{2}-12\left(10-2\sqrt{x}\right)=36$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(10-2\sqrt{x}\right)^{2} kengaytirilishi uchun ishlating.

$$36x-\left(100-40\sqrt{x}+4\left(\sqrt{x}\right)^{2}\right)-12\left(10-2\sqrt{x}\right)=36$$

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

$$36x-\left(100-40\sqrt{x}+4x\right)-12\left(10-2\sqrt{x}\right)=36$$

100-40\sqrt{x}+4x teskarisini topish uchun har birining teskarisini toping.

$$36x-100+40\sqrt{x}-4x-12\left(10-2\sqrt{x}\right)=36$$

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

$$32x-100+40\sqrt{x}-12\left(10-2\sqrt{x}\right)=36$$

-12 ga 10-2\sqrt{x} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$32x-100+40\sqrt{x}-120+24\sqrt{x}=36$$

-220 olish uchun -100 dan 120 ni ayirish.

$$32x-220+40\sqrt{x}+24\sqrt{x}=36$$

64\sqrt{x} ni olish uchun 40\sqrt{x} va 24\sqrt{x} ni birlashtirish.

$$32x-220+64\sqrt{x}=36$$

220 ni ikki tarafga qo’shing.

$$32x+64\sqrt{x}=36+220$$

256 olish uchun 36 va 220'ni qo'shing.

$$32x+64\sqrt{x}=256$$

Tenglamaning ikkala tarafidan 32x ni ayirish.

$$64\sqrt{x}=256-32x$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(64\sqrt{x}\right)^{2}=\left(-32x+256\right)^{2}$$

\left(64\sqrt{x}\right)^{2} ni kengaytirish.

$$64^{2}\left(\sqrt{x}\right)^{2}=\left(-32x+256\right)^{2}$$

2 daraja ko‘rsatkichini 64 ga hisoblang va 4096 ni qiymatni oling.

$$4096\left(\sqrt{x}\right)^{2}=\left(-32x+256\right)^{2}$$

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

$$4096x=\left(-32x+256\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(-32x+256\right)^{2} kengaytirilishi uchun ishlating.

$$4096x=1024x^{2}-16384x+65536$$

Ikkala tarafdan 1024x^{2} ni ayirish.

$$4096x-1024x^{2}=-16384x+65536$$

16384x ni ikki tarafga qo’shing.

$$4096x-1024x^{2}+16384x=65536$$

20480x ni olish uchun 4096x va 16384x ni birlashtirish.

$$20480x-1024x^{2}=65536$$

Ikkala tarafdan 65536 ni ayirish.

$$20480x-1024x^{2}-65536=0$$

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.

$$-1024x^{2}+20480x-65536=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1024 ni a, 20480 ni b va -65536 ni c bilan almashtiring.

$$x=\frac{-20480±\sqrt{20480^{2}-4\left(-1024\right)\left(-65536\right)}}{2\left(-1024\right)}$$

20480 kvadratini chiqarish.

$$x=\frac{-20480±\sqrt{419430400-4\left(-1024\right)\left(-65536\right)}}{2\left(-1024\right)}$$

-4 ni -1024 marotabaga ko'paytirish.

$$x=\frac{-20480±\sqrt{419430400+4096\left(-65536\right)}}{2\left(-1024\right)}$$

4096 ni -65536 marotabaga ko'paytirish.

$$x=\frac{-20480±\sqrt{419430400-268435456}}{2\left(-1024\right)}$$

419430400 ni -268435456 ga qo'shish.

$$x=\frac{-20480±\sqrt{150994944}}{2\left(-1024\right)}$$

150994944 ning kvadrat ildizini chiqarish.

$$x=\frac{-20480±12288}{2\left(-1024\right)}$$

2 ni -1024 marotabaga ko'paytirish.

$$x=\frac{-20480±12288}{-2048}$$

x=\frac{-20480±12288}{-2048} tenglamasini yeching, bunda ± musbat. -20480 ni 12288 ga qo'shish.

$$x=-\frac{8192}{-2048}$$

-8192 ni -2048 ga bo'lish.

$$x=4$$

x=\frac{-20480±12288}{-2048} tenglamasini yeching, bunda ± manfiy. -20480 dan 12288 ni ayirish.

$$x=-\frac{32768}{-2048}$$

-32768 ni -2048 ga bo'lish.

$$x=16$$

Tenglama yechildi.

$$x=4 x=16$$

2\sqrt{9x}-6=10-2\sqrt{x} tenglamasida x uchun 4 ni almashtiring.

$$2\sqrt{9\times 4}-6=10-2\sqrt{4}$$

Qisqartirish. x=4 tenglamani qoniqtiradi.

$$6=6$$

2\sqrt{9x}-6=10-2\sqrt{x} tenglamasida x uchun 16 ni almashtiring.

$$2\sqrt{9\times 16}-6=10-2\sqrt{16}$$

Qisqartirish. x=16 qiymati bu tenglamani qoniqtirmaydi.

$$18=2$$

2\sqrt{9x}-6=10-2\sqrt{x} tenglamasida x uchun 4 ni almashtiring.

$$2\sqrt{9\times 4}-6=10-2\sqrt{4}$$

Qisqartirish. x=4 tenglamani qoniqtiradi.

$$6=6$$

2\sqrt{9x}=10-2\sqrt{x}+6 tenglamasi noyob yechimga ega.

$$x=4$$