sqrt{x}+2x=2 eching

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

x uchun yechish

$$x=\frac{9-\sqrt{17}}{8}\approx 0,609611797$$

Yechim qadamlari

Tenglamaning ikkala tarafidan 2x ni ayirish.

$$\sqrt{x}=2-2x$$

Tenglamaning ikkala taraf kvadratini chiqarish.

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

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

$$x=\left(2-2x\right)^{2}$$

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

$$x=4-8x+4x^{2}$$

Ikkala tarafdan 4 ni ayirish.

$$x-4=-8x+4x^{2}$$

8x ni ikki tarafga qo’shing.

$$x-4+8x=4x^{2}$$

9x ni olish uchun x va 8x ni birlashtirish.

$$9x-4=4x^{2}$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$9x-4-4x^{2}=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-4x^{2}+9x-4=0$$

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

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

9 kvadratini chiqarish.

$$x=\frac{-9±\sqrt{81-4\left(-4\right)\left(-4\right)}}{2\left(-4\right)}$$

-4 ni -4 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81+16\left(-4\right)}}{2\left(-4\right)}$$

16 ni -4 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{81-64}}{2\left(-4\right)}$$

81 ni -64 ga qo'shish.

$$x=\frac{-9±\sqrt{17}}{2\left(-4\right)}$$

2 ni -4 marotabaga ko'paytirish.

$$x=\frac{-9±\sqrt{17}}{-8}$$

x=\frac{-9±\sqrt{17}}{-8} tenglamasini yeching, bunda ± musbat. -9 ni \sqrt{17} ga qo'shish.

$$x=\frac{\sqrt{17}-9}{-8}$$

-9+\sqrt{17} ni -8 ga bo'lish.

$$x=\frac{9-\sqrt{17}}{8}$$

x=\frac{-9±\sqrt{17}}{-8} tenglamasini yeching, bunda ± manfiy. -9 dan \sqrt{17} ni ayirish.

$$x=\frac{-\sqrt{17}-9}{-8}$$

-9-\sqrt{17} ni -8 ga bo'lish.

$$x=\frac{\sqrt{17}+9}{8}$$

Tenglama yechildi.

$$x=\frac{9-\sqrt{17}}{8} x=\frac{\sqrt{17}+9}{8}$$

\sqrt{x}+2x=2 tenglamasida x uchun \frac{9-\sqrt{17}}{8} ni almashtiring.

$$\sqrt{\frac{9-\sqrt{17}}{8}}+2\times \frac{9-\sqrt{17}}{8}=2$$

Qisqartirish. x=\frac{9-\sqrt{17}}{8} tenglamani qoniqtiradi.

$$2=2$$

\sqrt{x}+2x=2 tenglamasida x uchun \frac{\sqrt{17}+9}{8} ni almashtiring.

$$\sqrt{\frac{\sqrt{17}+9}{8}}+2\times \frac{\sqrt{17}+9}{8}=2$$

Qisqartirish. x=\frac{\sqrt{17}+9}{8} qiymati bu tenglamani qoniqtirmaydi.

$$\frac{5}{2}+\frac{1}{2}\times 17^{\frac{1}{2}}=2$$

\sqrt{x}=2-2x tenglamasi noyob yechimga ega.

$$x=\frac{9-\sqrt{17}}{8}$$