sqrt{4y+20}-sqrt{y-4}=6 eching

$$\sqrt { 4 y + 20 } - \sqrt { y - 4 } = 6$$

y uchun yechish

$$y=20$$

$$y=4$$

Yechim qadamlari

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

$$\sqrt{4y+20}=6+\sqrt{y-4}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{4y+20}\right)^{2}=\left(6+\sqrt{y-4}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{4y+20} ga hisoblang va 4y+20 ni qiymatni oling.

$$4y+20=\left(6+\sqrt{y-4}\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(6+\sqrt{y-4}\right)^{2} kengaytirilishi uchun ishlating.

$$4y+20=36+12\sqrt{y-4}+\left(\sqrt{y-4}\right)^{2}$$

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

$$4y+20=36+12\sqrt{y-4}+y-4$$

32 olish uchun 36 dan 4 ni ayirish.

$$4y+20=32+12\sqrt{y-4}+y$$

Tenglamaning ikkala tarafidan 32+y ni ayirish.

$$4y+20-\left(32+y\right)=12\sqrt{y-4}$$

32+y teskarisini topish uchun har birining teskarisini toping.

$$4y+20-32-y=12\sqrt{y-4}$$

-12 olish uchun 20 dan 32 ni ayirish.

$$4y-12-y=12\sqrt{y-4}$$

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

$$3y-12=12\sqrt{y-4}$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(3y-12\right)^{2}=\left(12\sqrt{y-4}\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(3y-12\right)^{2} kengaytirilishi uchun ishlating.

$$9y^{2}-72y+144=\left(12\sqrt{y-4}\right)^{2}$$

\left(12\sqrt{y-4}\right)^{2} ni kengaytirish.

$$9y^{2}-72y+144=12^{2}\left(\sqrt{y-4}\right)^{2}$$

2 daraja ko‘rsatkichini 12 ga hisoblang va 144 ni qiymatni oling.

$$9y^{2}-72y+144=144\left(\sqrt{y-4}\right)^{2}$$

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

$$9y^{2}-72y+144=144\left(y-4\right)$$

144 ga y-4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$9y^{2}-72y+144=144y-576$$

Ikkala tarafdan 144y ni ayirish.

$$9y^{2}-72y+144-144y=-576$$

-216y ni olish uchun -72y va -144y ni birlashtirish.

$$9y^{2}-216y+144=-576$$

576 ni ikki tarafga qo’shing.

$$9y^{2}-216y+144+576=0$$

720 olish uchun 144 va 576'ni qo'shing.

$$9y^{2}-216y+720=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 9 ni a, -216 ni b va 720 ni c bilan almashtiring.

$$y=\frac{-\left(-216\right)±\sqrt{\left(-216\right)^{2}-4\times 9\times 720}}{2\times 9}$$

-216 kvadratini chiqarish.

$$y=\frac{-\left(-216\right)±\sqrt{46656-4\times 9\times 720}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$y=\frac{-\left(-216\right)±\sqrt{46656-36\times 720}}{2\times 9}$$

-36 ni 720 marotabaga ko'paytirish.

$$y=\frac{-\left(-216\right)±\sqrt{46656-25920}}{2\times 9}$$

46656 ni -25920 ga qo'shish.

$$y=\frac{-\left(-216\right)±\sqrt{20736}}{2\times 9}$$

20736 ning kvadrat ildizini chiqarish.

$$y=\frac{-\left(-216\right)±144}{2\times 9}$$

-216 ning teskarisi 216 ga teng.

$$y=\frac{216±144}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$y=\frac{216±144}{18}$$

y=\frac{216±144}{18} tenglamasini yeching, bunda ± musbat. 216 ni 144 ga qo'shish.

$$y=\frac{360}{18}$$

360 ni 18 ga bo'lish.

$$y=20$$

y=\frac{216±144}{18} tenglamasini yeching, bunda ± manfiy. 216 dan 144 ni ayirish.

$$y=\frac{72}{18}$$

72 ni 18 ga bo'lish.

$$y=4$$

Tenglama yechildi.

$$y=20 y=4$$

\sqrt{4y+20}-\sqrt{y-4}=6 tenglamasida y uchun 20 ni almashtiring.

$$\sqrt{4\times 20+20}-\sqrt{20-4}=6$$

Qisqartirish. y=20 tenglamani qoniqtiradi.

$$6=6$$

\sqrt{4y+20}-\sqrt{y-4}=6 tenglamasida y uchun 4 ni almashtiring.

$$\sqrt{4\times 4+20}-\sqrt{4-4}=6$$

Qisqartirish. y=4 tenglamani qoniqtiradi.

$$6=6$$

\sqrt{4y+20}=\sqrt{y-4}+6 boʻyicha barcha yechimlar roʻyxati.

$$y=20 y=4$$