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

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

x uchun yechish (complex solution)

$$x=\frac{-16\sqrt{2}i-19}{9}\approx -2,111111111-2,514157444i$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

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

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

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

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

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

3 olish uchun -1 va 4'ni qo'shing.

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

\left(2\sqrt{x+3}\right)^{2} ni kengaytirish.

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

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

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

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

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

4 ga x+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x+3-4\sqrt{x-1}=4x+12$$

Tenglamaning ikkala tarafidan x+3 ni ayirish.

$$-4\sqrt{x-1}=4x+12-\left(x+3\right)$$

x+3 teskarisini topish uchun har birining teskarisini toping.

$$-4\sqrt{x-1}=4x+12-x-3$$

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

$$-4\sqrt{x-1}=3x+12-3$$

9 olish uchun 12 dan 3 ni ayirish.

$$-4\sqrt{x-1}=3x+9$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-4\sqrt{x-1}\right)^{2}=\left(3x+9\right)^{2}$$

\left(-4\sqrt{x-1}\right)^{2} ni kengaytirish.

$$\left(-4\right)^{2}\left(\sqrt{x-1}\right)^{2}=\left(3x+9\right)^{2}$$

2 daraja ko‘rsatkichini -4 ga hisoblang va 16 ni qiymatni oling.

$$16\left(\sqrt{x-1}\right)^{2}=\left(3x+9\right)^{2}$$

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

$$16\left(x-1\right)=\left(3x+9\right)^{2}$$

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

$$16x-16=\left(3x+9\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(3x+9\right)^{2} kengaytirilishi uchun ishlating.

$$16x-16=9x^{2}+54x+81$$

Ikkala tarafdan 9x^{2} ni ayirish.

$$16x-16-9x^{2}=54x+81$$

Ikkala tarafdan 54x ni ayirish.

$$16x-16-9x^{2}-54x=81$$

-38x ni olish uchun 16x va -54x ni birlashtirish.

$$-38x-16-9x^{2}=81$$

Ikkala tarafdan 81 ni ayirish.

$$-38x-16-9x^{2}-81=0$$

-97 olish uchun -16 dan 81 ni ayirish.

$$-38x-97-9x^{2}=0$$

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.

$$-9x^{2}-38x-97=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -9 ni a, -38 ni b va -97 ni c bilan almashtiring.

$$x=\frac{-\left(-38\right)±\sqrt{\left(-38\right)^{2}-4\left(-9\right)\left(-97\right)}}{2\left(-9\right)}$$

-38 kvadratini chiqarish.

$$x=\frac{-\left(-38\right)±\sqrt{1444-4\left(-9\right)\left(-97\right)}}{2\left(-9\right)}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-38\right)±\sqrt{1444+36\left(-97\right)}}{2\left(-9\right)}$$

36 ni -97 marotabaga ko'paytirish.

$$x=\frac{-\left(-38\right)±\sqrt{1444-3492}}{2\left(-9\right)}$$

1444 ni -3492 ga qo'shish.

$$x=\frac{-\left(-38\right)±\sqrt{-2048}}{2\left(-9\right)}$$

-2048 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-38\right)±32\sqrt{2}i}{2\left(-9\right)}$$

-38 ning teskarisi 38 ga teng.

$$x=\frac{38±32\sqrt{2}i}{2\left(-9\right)}$$

2 ni -9 marotabaga ko'paytirish.

$$x=\frac{38±32\sqrt{2}i}{-18}$$

x=\frac{38±32\sqrt{2}i}{-18} tenglamasini yeching, bunda ± musbat. 38 ni 32i\sqrt{2} ga qo'shish.

$$x=\frac{38+32\sqrt{2}i}{-18}$$

38+32i\sqrt{2} ni -18 ga bo'lish.

$$x=\frac{-16\sqrt{2}i-19}{9}$$

x=\frac{38±32\sqrt{2}i}{-18} tenglamasini yeching, bunda ± manfiy. 38 dan 32i\sqrt{2} ni ayirish.

$$x=\frac{-32\sqrt{2}i+38}{-18}$$

38-32i\sqrt{2} ni -18 ga bo'lish.

$$x=\frac{-19+16\sqrt{2}i}{9}$$

Tenglama yechildi.

$$x=\frac{-16\sqrt{2}i-19}{9} x=\frac{-19+16\sqrt{2}i}{9}$$

\sqrt{x-1}-2=2\sqrt{x+3} tenglamasida x uchun \frac{-16\sqrt{2}i-19}{9} ni almashtiring.

$$\sqrt{\frac{-16\sqrt{2}i-19}{9}-1}-2=2\sqrt{\frac{-16\sqrt{2}i-19}{9}+3}$$

Qisqartirish. x=\frac{-16\sqrt{2}i-19}{9} tenglamani qoniqtiradi.

$$-\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}=-\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}$$

\sqrt{x-1}-2=2\sqrt{x+3} tenglamasida x uchun \frac{-19+16\sqrt{2}i}{9} ni almashtiring.

$$\sqrt{\frac{-19+16\sqrt{2}i}{9}-1}-2=2\sqrt{\frac{-19+16\sqrt{2}i}{9}+3}$$

Qisqartirish. x=\frac{-19+16\sqrt{2}i}{9} qiymati bu tenglamani qoniqtirmaydi.

$$-\frac{4}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}=\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}$$

\sqrt{x-1}-2=2\sqrt{x+3} tenglamasida x uchun \frac{-16\sqrt{2}i-19}{9} ni almashtiring.

$$\sqrt{\frac{-16\sqrt{2}i-19}{9}-1}-2=2\sqrt{\frac{-16\sqrt{2}i-19}{9}+3}$$

Qisqartirish. x=\frac{-16\sqrt{2}i-19}{9} tenglamani qoniqtiradi.

$$-\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}=-\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}$$

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

$$x=\frac{-16\sqrt{2}i-19}{9}$$