x+3sqrt{x}+4=0 eching

$$x+3 \sqrt{ x } +4 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-3\sqrt{7}i+1}{2}\approx 0,5-3,968626967i$$

Yechim qadamlari

Tenglamaning ikkala tarafidan x+4 ni ayirish.

$$3\sqrt{x}=-\left(x+4\right)$$

x+4 teskarisini topish uchun har birining teskarisini toping.

$$3\sqrt{x}=-x-4$$

Tenglamaning ikkala taraf kvadratini chiqarish.

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

\left(3\sqrt{x}\right)^{2} ni kengaytirish.

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

2 daraja ko‘rsatkichini 3 ga hisoblang va 9 ni qiymatni oling.

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

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

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

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

$$9x=x^{2}+8x+16$$

Ikkala tarafdan x^{2} ni ayirish.

$$9x-x^{2}=8x+16$$

Ikkala tarafdan 8x ni ayirish.

$$9x-x^{2}-8x=16$$

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

$$x-x^{2}=16$$

Ikkala tarafdan 16 ni ayirish.

$$x-x^{2}-16=0$$

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.

$$-x^{2}+x-16=0$$

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 -16 ni c bilan almashtiring.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

1 kvadratini chiqarish.

$$x=\frac{-1±\sqrt{1-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1+4\left(-16\right)}}{2\left(-1\right)}$$

4 ni -16 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1-64}}{2\left(-1\right)}$$

1 ni -64 ga qo'shish.

$$x=\frac{-1±\sqrt{-63}}{2\left(-1\right)}$$

-63 ning kvadrat ildizini chiqarish.

$$x=\frac{-1±3\sqrt{7}i}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-1±3\sqrt{7}i}{-2}$$

x=\frac{-1±3\sqrt{7}i}{-2} tenglamasini yeching, bunda ± musbat. -1 ni 3i\sqrt{7} ga qo'shish.

$$x=\frac{-1+3\sqrt{7}i}{-2}$$

-1+3i\sqrt{7} ni -2 ga bo'lish.

$$x=\frac{-3\sqrt{7}i+1}{2}$$

x=\frac{-1±3\sqrt{7}i}{-2} tenglamasini yeching, bunda ± manfiy. -1 dan 3i\sqrt{7} ni ayirish.

$$x=\frac{-3\sqrt{7}i-1}{-2}$$

-1-3i\sqrt{7} ni -2 ga bo'lish.

$$x=\frac{1+3\sqrt{7}i}{2}$$

Tenglama yechildi.

$$x=\frac{-3\sqrt{7}i+1}{2} x=\frac{1+3\sqrt{7}i}{2}$$

x+3\sqrt{x}+4=0 tenglamasida x uchun \frac{-3\sqrt{7}i+1}{2} ni almashtiring.

$$\frac{-3\sqrt{7}i+1}{2}+3\sqrt{\frac{-3\sqrt{7}i+1}{2}}+4=0$$

Qisqartirish. x=\frac{-3\sqrt{7}i+1}{2} tenglamani qoniqtiradi.

$$0=0$$

x+3\sqrt{x}+4=0 tenglamasida x uchun \frac{1+3\sqrt{7}i}{2} ni almashtiring.

$$\frac{1+3\sqrt{7}i}{2}+3\sqrt{\frac{1+3\sqrt{7}i}{2}}+4=0$$

Qisqartirish. x=\frac{1+3\sqrt{7}i}{2} qiymati bu tenglamani qoniqtirmaydi.

$$9+3i\times 7^{\frac{1}{2}}=0$$

3\sqrt{x}=-x-4 tenglamasi noyob yechimga ega.

$$x=\frac{-3\sqrt{7}i+1}{2}$$