9(x+1)=sqrt{2x+5} eching

$$9 ( x + 1 ) = \sqrt { 2 x + 5 }$$

x uchun yechish

$$x=\frac{2\left(\sqrt{61}-40\right)}{81}\approx -0,79480865$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(9\left(x+1\right)\right)^{2}=\left(\sqrt{2x+5}\right)^{2}$$

9 ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

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

$$81x^{2}+162x+81=\left(\sqrt{2x+5}\right)^{2}$$

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

$$81x^{2}+162x+81=2x+5$$

Ikkala tarafdan 2x ni ayirish.

$$81x^{2}+162x+81-2x=5$$

160x ni olish uchun 162x va -2x ni birlashtirish.

$$81x^{2}+160x+81=5$$

Ikkala tarafdan 5 ni ayirish.

$$81x^{2}+160x+81-5=0$$

76 olish uchun 81 dan 5 ni ayirish.

$$81x^{2}+160x+76=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 81 ni a, 160 ni b va 76 ni c bilan almashtiring.

$$x=\frac{-160±\sqrt{160^{2}-4\times 81\times 76}}{2\times 81}$$

160 kvadratini chiqarish.

$$x=\frac{-160±\sqrt{25600-4\times 81\times 76}}{2\times 81}$$

-4 ni 81 marotabaga ko'paytirish.

$$x=\frac{-160±\sqrt{25600-324\times 76}}{2\times 81}$$

-324 ni 76 marotabaga ko'paytirish.

$$x=\frac{-160±\sqrt{25600-24624}}{2\times 81}$$

25600 ni -24624 ga qo'shish.

$$x=\frac{-160±\sqrt{976}}{2\times 81}$$

976 ning kvadrat ildizini chiqarish.

$$x=\frac{-160±4\sqrt{61}}{2\times 81}$$

2 ni 81 marotabaga ko'paytirish.

$$x=\frac{-160±4\sqrt{61}}{162}$$

x=\frac{-160±4\sqrt{61}}{162} tenglamasini yeching, bunda ± musbat. -160 ni 4\sqrt{61} ga qo'shish.

$$x=\frac{4\sqrt{61}-160}{162}$$

-160+4\sqrt{61} ni 162 ga bo'lish.

$$x=\frac{2\sqrt{61}-80}{81}$$

x=\frac{-160±4\sqrt{61}}{162} tenglamasini yeching, bunda ± manfiy. -160 dan 4\sqrt{61} ni ayirish.

$$x=\frac{-4\sqrt{61}-160}{162}$$

-160-4\sqrt{61} ni 162 ga bo'lish.

$$x=\frac{-2\sqrt{61}-80}{81}$$

Tenglama yechildi.

$$x=\frac{2\sqrt{61}-80}{81} x=\frac{-2\sqrt{61}-80}{81}$$

9\left(x+1\right)=\sqrt{2x+5} tenglamasida x uchun \frac{2\sqrt{61}-80}{81} ni almashtiring.

$$9\left(\frac{2\sqrt{61}-80}{81}+1\right)=\sqrt{2\times \frac{2\sqrt{61}-80}{81}+5}$$

Qisqartirish. x=\frac{2\sqrt{61}-80}{81} tenglamani qoniqtiradi.

$$\frac{2}{9}\times 61^{\frac{1}{2}}+\frac{1}{9}=\frac{2}{9}\times 61^{\frac{1}{2}}+\frac{1}{9}$$

9\left(x+1\right)=\sqrt{2x+5} tenglamasida x uchun \frac{-2\sqrt{61}-80}{81} ni almashtiring.

$$9\left(\frac{-2\sqrt{61}-80}{81}+1\right)=\sqrt{2\times \frac{-2\sqrt{61}-80}{81}+5}$$

Qisqartirish. x=\frac{-2\sqrt{61}-80}{81} qiymati bu tenglamani qoniqtirmaydi, chunki oʻng va chap tarafdagi belgilar bir-biriga qarama-qarshi.

$$-\frac{2}{9}\times 61^{\frac{1}{2}}+\frac{1}{9}=\frac{2}{9}\times 61^{\frac{1}{2}}-\frac{1}{9}$$

9\left(x+1\right)=\sqrt{2x+5} tenglamasi noyob yechimga ega.

$$x=\frac{2\sqrt{61}-80}{81}$$