sqrt{x+7}+sqrt{x+2}=sqrt{18x} eching

$$\sqrt { x + 7 } + \sqrt { x + 2 } = \sqrt { 18 x }$$

x uchun yechish

$$x = \frac{\sqrt{554} + 27}{42} \approx 1,203266776$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{x+7}+\sqrt{x+2}\right)^{2}=\left(\sqrt{18x}\right)^{2}$$

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

$$\left(\sqrt{x+7}\right)^{2}+2\sqrt{x+7}\sqrt{x+2}+\left(\sqrt{x+2}\right)^{2}=\left(\sqrt{18x}\right)^{2}$$

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

$$x+7+2\sqrt{x+7}\sqrt{x+2}+\left(\sqrt{x+2}\right)^{2}=\left(\sqrt{18x}\right)^{2}$$

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

$$x+7+2\sqrt{x+7}\sqrt{x+2}+x+2=\left(\sqrt{18x}\right)^{2}$$

2x ni olish uchun x va x ni birlashtirish.

$$2x+7+2\sqrt{x+7}\sqrt{x+2}+2=\left(\sqrt{18x}\right)^{2}$$

9 olish uchun 7 va 2'ni qo'shing.

$$2x+9+2\sqrt{x+7}\sqrt{x+2}=\left(\sqrt{18x}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{18x} ga hisoblang va 18x ni qiymatni oling.

$$2x+9+2\sqrt{x+7}\sqrt{x+2}=18x$$

Tenglamaning ikkala tarafidan 2x+9 ni ayirish.

$$2\sqrt{x+7}\sqrt{x+2}=18x-\left(2x+9\right)$$

2x+9 teskarisini topish uchun har birining teskarisini toping.

$$2\sqrt{x+7}\sqrt{x+2}=18x-2x-9$$

16x ni olish uchun 18x va -2x ni birlashtirish.

$$2\sqrt{x+7}\sqrt{x+2}=16x-9$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(2\sqrt{x+7}\sqrt{x+2}\right)^{2}=\left(16x-9\right)^{2}$$

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

$$2^{2}\left(\sqrt{x+7}\right)^{2}\left(\sqrt{x+2}\right)^{2}=\left(16x-9\right)^{2}$$

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

$$4\left(\sqrt{x+7}\right)^{2}\left(\sqrt{x+2}\right)^{2}=\left(16x-9\right)^{2}$$

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

$$4\left(x+7\right)\left(\sqrt{x+2}\right)^{2}=\left(16x-9\right)^{2}$$

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

$$4\left(x+7\right)\left(x+2\right)=\left(16x-9\right)^{2}$$

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

$$\left(4x+28\right)\left(x+2\right)=\left(16x-9\right)^{2}$$

4x+28 ifodaning har bir elementini x+2 ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$4x^{2}+8x+28x+56=\left(16x-9\right)^{2}$$

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

$$4x^{2}+36x+56=\left(16x-9\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(16x-9\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}+36x+56=256x^{2}-288x+81$$

Ikkala tarafdan 256x^{2} ni ayirish.

$$4x^{2}+36x+56-256x^{2}=-288x+81$$

-252x^{2} ni olish uchun 4x^{2} va -256x^{2} ni birlashtirish.

$$-252x^{2}+36x+56=-288x+81$$

288x ni ikki tarafga qo’shing.

$$-252x^{2}+36x+56+288x=81$$

324x ni olish uchun 36x va 288x ni birlashtirish.

$$-252x^{2}+324x+56=81$$

Ikkala tarafdan 81 ni ayirish.

$$-252x^{2}+324x+56-81=0$$

-25 olish uchun 56 dan 81 ni ayirish.

$$-252x^{2}+324x-25=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -252 ni a, 324 ni b va -25 ni c bilan almashtiring.

$$x=\frac{-324±\sqrt{324^{2}-4\left(-252\right)\left(-25\right)}}{2\left(-252\right)}$$

324 kvadratini chiqarish.

$$x=\frac{-324±\sqrt{104976-4\left(-252\right)\left(-25\right)}}{2\left(-252\right)}$$

-4 ni -252 marotabaga ko'paytirish.

$$x=\frac{-324±\sqrt{104976+1008\left(-25\right)}}{2\left(-252\right)}$$

1008 ni -25 marotabaga ko'paytirish.

$$x=\frac{-324±\sqrt{104976-25200}}{2\left(-252\right)}$$

104976 ni -25200 ga qo'shish.

$$x=\frac{-324±\sqrt{79776}}{2\left(-252\right)}$$

79776 ning kvadrat ildizini chiqarish.

$$x=\frac{-324±12\sqrt{554}}{2\left(-252\right)}$$

2 ni -252 marotabaga ko'paytirish.

$$x=\frac{-324±12\sqrt{554}}{-504}$$

x=\frac{-324±12\sqrt{554}}{-504} tenglamasini yeching, bunda ± musbat. -324 ni 12\sqrt{554} ga qo'shish.

$$x=\frac{12\sqrt{554}-324}{-504}$$

-324+12\sqrt{554} ni -504 ga bo'lish.

$$x=-\frac{\sqrt{554}}{42}+\frac{9}{14}$$

x=\frac{-324±12\sqrt{554}}{-504} tenglamasini yeching, bunda ± manfiy. -324 dan 12\sqrt{554} ni ayirish.

$$x=\frac{-12\sqrt{554}-324}{-504}$$

-324-12\sqrt{554} ni -504 ga bo'lish.

$$x=\frac{\sqrt{554}}{42}+\frac{9}{14}$$

Tenglama yechildi.

$$x=-\frac{\sqrt{554}}{42}+\frac{9}{14} x=\frac{\sqrt{554}}{42}+\frac{9}{14}$$

\sqrt{x+7}+\sqrt{x+2}=\sqrt{18x} tenglamasida x uchun -\frac{\sqrt{554}}{42}+\frac{9}{14} ni almashtiring.

$$\sqrt{-\frac{\sqrt{554}}{42}+\frac{9}{14}+7}+\sqrt{-\frac{\sqrt{554}}{42}+\frac{9}{14}+2}=\sqrt{18\left(-\frac{\sqrt{554}}{42}+\frac{9}{14}\right)}$$

Qisqartirish. x=-\frac{\sqrt{554}}{42}+\frac{9}{14} qiymati bu tenglamani qoniqtirmaydi.

$$\left(-\frac{1}{42}\times 554^{\frac{1}{2}}+\frac{107}{14}\right)^{\frac{1}{2}}+\left(-\frac{1}{42}\times 554^{\frac{1}{2}}+\frac{37}{14}\right)^{\frac{1}{2}}=\left(-\frac{3}{7}\times 554^{\frac{1}{2}}+\frac{81}{7}\right)^{\frac{1}{2}}$$

\sqrt{x+7}+\sqrt{x+2}=\sqrt{18x} tenglamasida x uchun \frac{\sqrt{554}}{42}+\frac{9}{14} ni almashtiring.

$$\sqrt{\frac{\sqrt{554}}{42}+\frac{9}{14}+7}+\sqrt{\frac{\sqrt{554}}{42}+\frac{9}{14}+2}=\sqrt{18\left(\frac{\sqrt{554}}{42}+\frac{9}{14}\right)}$$

Qisqartirish. x=\frac{\sqrt{554}}{42}+\frac{9}{14} tenglamani qoniqtiradi.

$$\left(\frac{1}{42}\times 554^{\frac{1}{2}}+\frac{107}{14}\right)^{\frac{1}{2}}+\left(\frac{1}{42}\times 554^{\frac{1}{2}}+\frac{37}{14}\right)^{\frac{1}{2}}=\left(\frac{3}{7}\times 554^{\frac{1}{2}}+\frac{81}{7}\right)^{\frac{1}{2}}$$

\sqrt{x+2}+\sqrt{x+7}=\sqrt{18x} tenglamasi noyob yechimga ega.

$$x=\frac{\sqrt{554}}{42}+\frac{9}{14}$$