sqrt{x}-x=1 eching

$$\sqrt{ x } -x=1$$

x uchun yechish (complex solution)

$$x=\frac{-1+\sqrt{3}i}{2}\approx -0,5+0,866025404i$$

Yechim qadamlari

Tenglamaning ikkala tarafidan -x ni ayirish.

$$\sqrt{x}=1+x$$

Tenglamaning ikkala taraf kvadratini chiqarish.

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

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

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(1+x\right)^{2} kengaytirilishi uchun ishlating.

$$x=1+2x+x^{2}$$

Ikkala tarafdan 1 ni ayirish.

$$x-1=2x+x^{2}$$

Ikkala tarafdan 2x ni ayirish.

$$x-1-2x=x^{2}$$

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

$$-x-1=x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$-x-1-x^{2}=0$$

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.

$$-x^{2}-x-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -1 ni b va -1 ni c bilan almashtiring.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+4\left(-1\right)}}{2\left(-1\right)}$$

4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-4}}{2\left(-1\right)}$$

1 ni -4 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{-3}}{2\left(-1\right)}$$

-3 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1\right)±\sqrt{3}i}{2\left(-1\right)}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{3}i}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{3}i}{-2}$$

x=\frac{1±\sqrt{3}i}{-2} tenglamasini yeching, bunda ± musbat. 1 ni i\sqrt{3} ga qo'shish.

$$x=\frac{1+\sqrt{3}i}{-2}$$

1+i\sqrt{3} ni -2 ga bo'lish.

$$x=\frac{-\sqrt{3}i-1}{2}$$

x=\frac{1±\sqrt{3}i}{-2} tenglamasini yeching, bunda ± manfiy. 1 dan i\sqrt{3} ni ayirish.

$$x=\frac{-\sqrt{3}i+1}{-2}$$

1-i\sqrt{3} ni -2 ga bo'lish.

$$x=\frac{-1+\sqrt{3}i}{2}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{3}i-1}{2} x=\frac{-1+\sqrt{3}i}{2}$$

\sqrt{x}-x=1 tenglamasida x uchun \frac{-\sqrt{3}i-1}{2} ni almashtiring.

$$\sqrt{\frac{-\sqrt{3}i-1}{2}}-\frac{-\sqrt{3}i-1}{2}=1$$

Qisqartirish. x=\frac{-\sqrt{3}i-1}{2} qiymati bu tenglamani qoniqtirmaydi.

$$i\times 3^{\frac{1}{2}}=1$$

\sqrt{x}-x=1 tenglamasida x uchun \frac{-1+\sqrt{3}i}{2} ni almashtiring.

$$\sqrt{\frac{-1+\sqrt{3}i}{2}}-\frac{-1+\sqrt{3}i}{2}=1$$

Qisqartirish. x=\frac{-1+\sqrt{3}i}{2} tenglamani qoniqtiradi.

$$1=1$$

\sqrt{x}=x+1 tenglamasi noyob yechimga ega.

$$x=\frac{-1+\sqrt{3}i}{2}$$