3sqrt{2x-3}+2sqrt{7-x}=11 eching

$$3 \sqrt{ 2x-3 } +2 \sqrt{ 7-x } =11$$

x uchun yechish

$$x=6$$

Yechim qadamlari

Tenglamaning ikkala tarafidan 2\sqrt{7-x} ni ayirish.

$$3\sqrt{2x-3}=11-2\sqrt{7-x}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(3\sqrt{2x-3}\right)^{2}=\left(11-2\sqrt{7-x}\right)^{2}$$

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

$$3^{2}\left(\sqrt{2x-3}\right)^{2}=\left(11-2\sqrt{7-x}\right)^{2}$$

2 daraja ko‘rsatkichini 3 ga hisoblang va 9 ni qiymatni oling.

$$9\left(\sqrt{2x-3}\right)^{2}=\left(11-2\sqrt{7-x}\right)^{2}$$

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

$$9\left(2x-3\right)=\left(11-2\sqrt{7-x}\right)^{2}$$

9 ga 2x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$18x-27=\left(11-2\sqrt{7-x}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(11-2\sqrt{7-x}\right)^{2} kengaytirilishi uchun ishlating.

$$18x-27=121-44\sqrt{7-x}+4\left(\sqrt{7-x}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{7-x} ga hisoblang va 7-x ni qiymatni oling.

$$18x-27=121-44\sqrt{7-x}+4\left(7-x\right)$$

4 ga 7-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$18x-27=121-44\sqrt{7-x}+28-4x$$

149 olish uchun 121 va 28'ni qo'shing.

$$18x-27=149-44\sqrt{7-x}-4x$$

Tenglamaning ikkala tarafidan 149-4x ni ayirish.

$$18x-27-\left(149-4x\right)=-44\sqrt{7-x}$$

149-4x teskarisini topish uchun har birining teskarisini toping.

$$18x-27-149+4x=-44\sqrt{7-x}$$

-176 olish uchun -27 dan 149 ni ayirish.

$$18x-176+4x=-44\sqrt{7-x}$$

22x ni olish uchun 18x va 4x ni birlashtirish.

$$22x-176=-44\sqrt{7-x}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(22x-176\right)^{2}=\left(-44\sqrt{7-x}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(22x-176\right)^{2} kengaytirilishi uchun ishlating.

$$484x^{2}-7744x+30976=\left(-44\sqrt{7-x}\right)^{2}$$

\left(-44\sqrt{7-x}\right)^{2} ni kengaytirish.

$$484x^{2}-7744x+30976=\left(-44\right)^{2}\left(\sqrt{7-x}\right)^{2}$$

2 daraja ko‘rsatkichini -44 ga hisoblang va 1936 ni qiymatni oling.

$$484x^{2}-7744x+30976=1936\left(\sqrt{7-x}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{7-x} ga hisoblang va 7-x ni qiymatni oling.

$$484x^{2}-7744x+30976=1936\left(7-x\right)$$

1936 ga 7-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$484x^{2}-7744x+30976=13552-1936x$$

Ikkala tarafdan 13552 ni ayirish.

$$484x^{2}-7744x+30976-13552=-1936x$$

17424 olish uchun 30976 dan 13552 ni ayirish.

$$484x^{2}-7744x+17424=-1936x$$

1936x ni ikki tarafga qo’shing.

$$484x^{2}-7744x+17424+1936x=0$$

-5808x ni olish uchun -7744x va 1936x ni birlashtirish.

$$484x^{2}-5808x+17424=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 484 ni a, -5808 ni b va 17424 ni c bilan almashtiring.

$$x=\frac{-\left(-5808\right)±\sqrt{\left(-5808\right)^{2}-4\times 484\times 17424}}{2\times 484}$$

-5808 kvadratini chiqarish.

$$x=\frac{-\left(-5808\right)±\sqrt{33732864-4\times 484\times 17424}}{2\times 484}$$

-4 ni 484 marotabaga ko'paytirish.

$$x=\frac{-\left(-5808\right)±\sqrt{33732864-1936\times 17424}}{2\times 484}$$

-1936 ni 17424 marotabaga ko'paytirish.

$$x=\frac{-\left(-5808\right)±\sqrt{33732864-33732864}}{2\times 484}$$

33732864 ni -33732864 ga qo'shish.

$$x=\frac{-\left(-5808\right)±\sqrt{0}}{2\times 484}$$

0 ning kvadrat ildizini chiqarish.

$$x=-\frac{-5808}{2\times 484}$$

-5808 ning teskarisi 5808 ga teng.

$$x=\frac{5808}{2\times 484}$$

2 ni 484 marotabaga ko'paytirish.

$$x=\frac{5808}{968}$$

5808 ni 968 ga bo'lish.

$$x=6$$

3\sqrt{2x-3}+2\sqrt{7-x}=11 tenglamasida x uchun 6 ni almashtiring.

$$3\sqrt{2\times 6-3}+2\sqrt{7-6}=11$$

Qisqartirish. x=6 tenglamani qoniqtiradi.

$$11=11$$

3\sqrt{2x-3}=-2\sqrt{7-x}+11 tenglamasi noyob yechimga ega.

$$x=6$$