sqrt{2x+9}-sqrt{x-4}-3=0 eching

$$\sqrt { 2 x + 9 } - \sqrt { x - 4 } - 3 = 0$$

x uchun yechish

$$x=20$$

$$x=8$$

Yechim qadamlari

Tenglamaning ikkala tarafidan -\sqrt{x-4}-3 ni ayirish.

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

-\sqrt{x-4}-3 teskarisini topish uchun har birining teskarisini toping.

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

-\sqrt{x-4} ning teskarisi \sqrt{x-4} ga teng.

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

-3 ning teskarisi 3 ga teng.

$$\sqrt{2x+9}=\sqrt{x-4}+3$$

Tenglamaning ikkala taraf kvadratini chiqarish.

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

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

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(\sqrt{x-4}+3\right)^{2} kengaytirilishi uchun ishlating.

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

2 daraja ko‘rsatkichini \sqrt{x-4} ga hisoblang va x-4 ni qiymatni oling.

$$2x+9=x-4+6\sqrt{x-4}+9$$

5 olish uchun -4 va 9'ni qo'shing.

$$2x+9=x+5+6\sqrt{x-4}$$

Tenglamaning ikkala tarafidan x+5 ni ayirish.

$$2x+9-\left(x+5\right)=6\sqrt{x-4}$$

x+5 teskarisini topish uchun har birining teskarisini toping.

$$2x+9-x-5=6\sqrt{x-4}$$

x ni olish uchun 2x va -x ni birlashtirish.

$$x+9-5=6\sqrt{x-4}$$

4 olish uchun 9 dan 5 ni ayirish.

$$x+4=6\sqrt{x-4}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+4\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+8x+16=\left(6\sqrt{x-4}\right)^{2}$$

\left(6\sqrt{x-4}\right)^{2} ni kengaytirish.

$$x^{2}+8x+16=6^{2}\left(\sqrt{x-4}\right)^{2}$$

2 daraja ko‘rsatkichini 6 ga hisoblang va 36 ni qiymatni oling.

$$x^{2}+8x+16=36\left(\sqrt{x-4}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{x-4} ga hisoblang va x-4 ni qiymatni oling.

$$x^{2}+8x+16=36\left(x-4\right)$$

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

$$x^{2}+8x+16=36x-144$$

Ikkala tarafdan 36x ni ayirish.

$$x^{2}+8x+16-36x=-144$$

-28x ni olish uchun 8x va -36x ni birlashtirish.

$$x^{2}-28x+16=-144$$

144 ni ikki tarafga qo’shing.

$$x^{2}-28x+16+144=0$$

160 olish uchun 16 va 144'ni qo'shing.

$$x^{2}-28x+160=0$$

Bu tenglamani yechish uchun x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right) formulasi yordamida x^{2}-28x+160 ni faktorlang. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-28 ab=160$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 160-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-160 -2,-80 -4,-40 -5,-32 -8,-20 -10,-16$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-160=-161 -2-80=-82 -4-40=-44 -5-32=-37 -8-20=-28 -10-16=-26$$

Yechim – -28 yigʻindisini beruvchi juftlik.

$$a=-20 b=-8$$

Faktorlangan \left(x+a\right)\left(x+b\right) ifodani olingan qiymatlar bilan qaytadan yozing.

$$\left(x-20\right)\left(x-8\right)$$

Tenglamani yechish uchun x-20=0 va x-8=0 ni yeching.

$$x=20 x=8$$

\sqrt{2x+9}-\sqrt{x-4}-3=0 tenglamasida x uchun 20 ni almashtiring.

$$\sqrt{2\times 20+9}-\sqrt{20-4}-3=0$$

Qisqartirish. x=20 tenglamani qoniqtiradi.

$$0=0$$

\sqrt{2x+9}-\sqrt{x-4}-3=0 tenglamasida x uchun 8 ni almashtiring.

$$\sqrt{2\times 8+9}-\sqrt{8-4}-3=0$$

Qisqartirish. x=8 tenglamani qoniqtiradi.

$$0=0$$

\sqrt{2x+9}=\sqrt{x-4}+3 boʻyicha barcha yechimlar roʻyxati.

$$x=20 x=8$$