sqrt{a^2-4a+20}=sqrt{a} eching

$$\sqrt { a ^ { 2 } - 4 a + 20 } = \sqrt { a }$$

a uchun yechish

$$a=2\sqrt{5}e^{\arctan(\frac{\sqrt{55}}{5})i}\approx 2,5+3,708099244i$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{a^{2}-4a+20}\right)^{2}=\left(\sqrt{a}\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{a^{2}-4a+20} ga hisoblang va a^{2}-4a+20 ni qiymatni oling.

$$a^{2}-4a+20=\left(\sqrt{a}\right)^{2}$$

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

$$a^{2}-4a+20=a$$

Ikkala tarafdan a ni ayirish.

$$a^{2}-4a+20-a=0$$

-5a ni olish uchun -4a va -a ni birlashtirish.

$$a^{2}-5a+20=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -5 ni b va 20 ni c bilan almashtiring.

$$a=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 20}}{2}$$

-5 kvadratini chiqarish.

$$a=\frac{-\left(-5\right)±\sqrt{25-4\times 20}}{2}$$

-4 ni 20 marotabaga ko'paytirish.

$$a=\frac{-\left(-5\right)±\sqrt{25-80}}{2}$$

25 ni -80 ga qo'shish.

$$a=\frac{-\left(-5\right)±\sqrt{-55}}{2}$$

-55 ning kvadrat ildizini chiqarish.

$$a=\frac{-\left(-5\right)±\sqrt{55}i}{2}$$

-5 ning teskarisi 5 ga teng.

$$a=\frac{5±\sqrt{55}i}{2}$$

a=\frac{5±\sqrt{55}i}{2} tenglamasini yeching, bunda ± musbat. 5 ni i\sqrt{55} ga qo'shish.

$$a=\frac{5+\sqrt{55}i}{2}$$

a=\frac{5±\sqrt{55}i}{2} tenglamasini yeching, bunda ± manfiy. 5 dan i\sqrt{55} ni ayirish.

$$a=\frac{-\sqrt{55}i+5}{2}$$

Tenglama yechildi.

$$a=\frac{5+\sqrt{55}i}{2} a=\frac{-\sqrt{55}i+5}{2}$$

\sqrt{a^{2}-4a+20}=\sqrt{a} tenglamasida a uchun \frac{5+\sqrt{55}i}{2} ni almashtiring.

$$\sqrt{\left(\frac{5+\sqrt{55}i}{2}\right)^{2}-4\times \frac{5+\sqrt{55}i}{2}+20}=\sqrt{\frac{5+\sqrt{55}i}{2}}$$

Qisqartirish. a=\frac{5+\sqrt{55}i}{2} tenglamani qoniqtiradi.

$$\frac{1}{2}\left(10+2i\times 55^{\frac{1}{2}}\right)^{\frac{1}{2}}=\left(\frac{5}{2}+\frac{1}{2}i\times 55^{\frac{1}{2}}\right)^{\frac{1}{2}}$$

\sqrt{a^{2}-4a+20}=\sqrt{a} tenglamasida a uchun \frac{-\sqrt{55}i+5}{2} ni almashtiring.

$$\sqrt{\left(\frac{-\sqrt{55}i+5}{2}\right)^{2}-4\times \frac{-\sqrt{55}i+5}{2}+20}=\sqrt{\frac{-\sqrt{55}i+5}{2}}$$

Qisqartirish. a=\frac{-\sqrt{55}i+5}{2} tenglamani qoniqtiradi.

$$\frac{1}{2}\left(10-2i\times 55^{\frac{1}{2}}\right)^{\frac{1}{2}}=\left(-\frac{1}{2}i\times 55^{\frac{1}{2}}+\frac{5}{2}\right)^{\frac{1}{2}}$$

\sqrt{a^{2}-4a+20}=\sqrt{a} boʻyicha barcha yechimlar roʻyxati.

$$a=\frac{5+\sqrt{55}i}{2} a=\frac{-\sqrt{55}i+5}{2}$$