sqrt{x+3}=x eching

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

x uchun yechish

$$x = \frac{\sqrt{13} + 1}{2} \approx 2,302775638$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

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

2 daraja ko‘rsatkichini \sqrt{x+3} ga hisoblang va x+3 ni qiymatni oling.

$$x+3=x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$x+3-x^{2}=0$$

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.

$$-x^{2}+x+3=0$$

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 3 ni c bilan almashtiring.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

1 kvadratini chiqarish.

$$x=\frac{-1±\sqrt{1-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1+4\times 3}}{2\left(-1\right)}$$

4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1+12}}{2\left(-1\right)}$$

1 ni 12 ga qo'shish.

$$x=\frac{-1±\sqrt{13}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{13}}{-2}$$

x=\frac{-1±\sqrt{13}}{-2} tenglamasini yeching, bunda ± musbat. -1 ni \sqrt{13} ga qo'shish.

$$x=\frac{\sqrt{13}-1}{-2}$$

-1+\sqrt{13} ni -2 ga bo'lish.

$$x=\frac{1-\sqrt{13}}{2}$$

x=\frac{-1±\sqrt{13}}{-2} tenglamasini yeching, bunda ± manfiy. -1 dan \sqrt{13} ni ayirish.

$$x=\frac{-\sqrt{13}-1}{-2}$$

-1-\sqrt{13} ni -2 ga bo'lish.

$$x=\frac{\sqrt{13}+1}{2}$$

Tenglama yechildi.

$$x=\frac{1-\sqrt{13}}{2} x=\frac{\sqrt{13}+1}{2}$$

\sqrt{x+3}=x tenglamasida x uchun \frac{1-\sqrt{13}}{2} ni almashtiring.

$$\sqrt{\frac{1-\sqrt{13}}{2}+3}=\frac{1-\sqrt{13}}{2}$$

Qisqartirish. x=\frac{1-\sqrt{13}}{2} qiymati bu tenglamani qoniqtirmaydi, chunki oʻng va chap tarafdagi belgilar bir-biriga qarama-qarshi.

$$-\left(\frac{1}{2}-\frac{1}{2}\times 13^{\frac{1}{2}}\right)=\frac{1}{2}-\frac{1}{2}\times 13^{\frac{1}{2}}$$

\sqrt{x+3}=x tenglamasida x uchun \frac{\sqrt{13}+1}{2} ni almashtiring.

$$\sqrt{\frac{\sqrt{13}+1}{2}+3}=\frac{\sqrt{13}+1}{2}$$

Qisqartirish. x=\frac{\sqrt{13}+1}{2} tenglamani qoniqtiradi.

$$\frac{1}{2}+\frac{1}{2}\times 13^{\frac{1}{2}}=\frac{1}{2}\times 13^{\frac{1}{2}}+\frac{1}{2}$$

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

$$x=\frac{\sqrt{13}+1}{2}$$