sqrt{x-5}+3sqrt{x+3}=10 eching

$$\sqrt { x - 5 } + 3 \sqrt { x + 3 } = 10$$

x uchun yechish

$$x=6$$

Yechim qadamlari

Tenglamaning ikkala tarafidan 3\sqrt{x+3} ni ayirish.

$$\sqrt{x-5}=10-3\sqrt{x+3}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{x-5}\right)^{2}=\left(10-3\sqrt{x+3}\right)^{2}$$

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

$$x-5=\left(10-3\sqrt{x+3}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(10-3\sqrt{x+3}\right)^{2} kengaytirilishi uchun ishlating.

$$x-5=100-60\sqrt{x+3}+9\left(\sqrt{x+3}\right)^{2}$$

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

$$x-5=100-60\sqrt{x+3}+9\left(x+3\right)$$

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

$$x-5=100-60\sqrt{x+3}+9x+27$$

127 olish uchun 100 va 27'ni qo'shing.

$$x-5=127-60\sqrt{x+3}+9x$$

Tenglamaning ikkala tarafidan 127+9x ni ayirish.

$$x-5-\left(127+9x\right)=-60\sqrt{x+3}$$

127+9x teskarisini topish uchun har birining teskarisini toping.

$$x-5-127-9x=-60\sqrt{x+3}$$

-132 olish uchun -5 dan 127 ni ayirish.

$$x-132-9x=-60\sqrt{x+3}$$

-8x ni olish uchun x va -9x ni birlashtirish.

$$-8x-132=-60\sqrt{x+3}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(-8x-132\right)^{2}=\left(-60\sqrt{x+3}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(-8x-132\right)^{2} kengaytirilishi uchun ishlating.

$$64x^{2}+2112x+17424=\left(-60\sqrt{x+3}\right)^{2}$$

\left(-60\sqrt{x+3}\right)^{2} ni kengaytirish.

$$64x^{2}+2112x+17424=\left(-60\right)^{2}\left(\sqrt{x+3}\right)^{2}$$

2 daraja ko‘rsatkichini -60 ga hisoblang va 3600 ni qiymatni oling.

$$64x^{2}+2112x+17424=3600\left(\sqrt{x+3}\right)^{2}$$

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

$$64x^{2}+2112x+17424=3600\left(x+3\right)$$

3600 ga x+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$64x^{2}+2112x+17424=3600x+10800$$

Ikkala tarafdan 3600x ni ayirish.

$$64x^{2}+2112x+17424-3600x=10800$$

-1488x ni olish uchun 2112x va -3600x ni birlashtirish.

$$64x^{2}-1488x+17424=10800$$

Ikkala tarafdan 10800 ni ayirish.

$$64x^{2}-1488x+17424-10800=0$$

6624 olish uchun 17424 dan 10800 ni ayirish.

$$64x^{2}-1488x+6624=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 64 ni a, -1488 ni b va 6624 ni c bilan almashtiring.

$$x=\frac{-\left(-1488\right)±\sqrt{\left(-1488\right)^{2}-4\times 64\times 6624}}{2\times 64}$$

-1488 kvadratini chiqarish.

$$x=\frac{-\left(-1488\right)±\sqrt{2214144-4\times 64\times 6624}}{2\times 64}$$

-4 ni 64 marotabaga ko'paytirish.

$$x=\frac{-\left(-1488\right)±\sqrt{2214144-256\times 6624}}{2\times 64}$$

-256 ni 6624 marotabaga ko'paytirish.

$$x=\frac{-\left(-1488\right)±\sqrt{2214144-1695744}}{2\times 64}$$

2214144 ni -1695744 ga qo'shish.

$$x=\frac{-\left(-1488\right)±\sqrt{518400}}{2\times 64}$$

518400 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1488\right)±720}{2\times 64}$$

-1488 ning teskarisi 1488 ga teng.

$$x=\frac{1488±720}{2\times 64}$$

2 ni 64 marotabaga ko'paytirish.

$$x=\frac{1488±720}{128}$$

x=\frac{1488±720}{128} tenglamasini yeching, bunda ± musbat. 1488 ni 720 ga qo'shish.

$$x=\frac{2208}{128}$$

\frac{2208}{128} ulushini 32 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{69}{4}$$

x=\frac{1488±720}{128} tenglamasini yeching, bunda ± manfiy. 1488 dan 720 ni ayirish.

$$x=\frac{768}{128}$$

768 ni 128 ga bo'lish.

$$x=6$$

Tenglama yechildi.

$$x=\frac{69}{4} x=6$$

\sqrt{x-5}+3\sqrt{x+3}=10 tenglamasida x uchun \frac{69}{4} ni almashtiring.

$$\sqrt{\frac{69}{4}-5}+3\sqrt{\frac{69}{4}+3}=10$$

Qisqartirish. x=\frac{69}{4} qiymati bu tenglamani qoniqtirmaydi.

$$17=10$$

\sqrt{x-5}+3\sqrt{x+3}=10 tenglamasida x uchun 6 ni almashtiring.

$$\sqrt{6-5}+3\sqrt{6+3}=10$$

Qisqartirish. x=6 tenglamani qoniqtiradi.

$$10=10$$

\sqrt{x-5}=-3\sqrt{x+3}+10 tenglamasi noyob yechimga ega.

$$x=6$$