x^2+(x-6)^2=16 eching

$$x ^ { 2 } + ( x - 6 ) ^ { 2 } = 16$$

x uchun yechish (complex solution)

$$x=3+i$$

$$x=3-i$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-6\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+x^{2}-12x+36=16$$

2x^{2} ni olish uchun x^{2} va x^{2} ni birlashtirish.

$$2x^{2}-12x+36=16$$

Ikkala tarafdan 16 ni ayirish.

$$2x^{2}-12x+36-16=0$$

20 olish uchun 36 dan 16 ni ayirish.

$$2x^{2}-12x+20=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -12 ni b va 20 ni c bilan almashtiring.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 2\times 20}}{2\times 2}$$

-12 kvadratini chiqarish.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 2\times 20}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-8\times 20}}{2\times 2}$$

-8 ni 20 marotabaga ko'paytirish.

$$x=\frac{-\left(-12\right)±\sqrt{144-160}}{2\times 2}$$

144 ni -160 ga qo'shish.

$$x=\frac{-\left(-12\right)±\sqrt{-16}}{2\times 2}$$

-16 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-12\right)±4i}{2\times 2}$$

-12 ning teskarisi 12 ga teng.

$$x=\frac{12±4i}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{12±4i}{4}$$

x=\frac{12±4i}{4} tenglamasini yeching, bunda ± musbat. 12 ni 4i ga qo'shish.

$$x=\frac{12+4i}{4}$$

12+4i ni 4 ga bo'lish.

$$x=3+i$$

x=\frac{12±4i}{4} tenglamasini yeching, bunda ± manfiy. 12 dan 4i ni ayirish.

$$x=\frac{12-4i}{4}$$

12-4i ni 4 ga bo'lish.

$$x=3-i$$

Tenglama yechildi.

$$x=3+i x=3-i$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-6\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+x^{2}-12x+36=16$$

2x^{2} ni olish uchun x^{2} va x^{2} ni birlashtirish.

$$2x^{2}-12x+36=16$$

Ikkala tarafdan 36 ni ayirish.

$$2x^{2}-12x=16-36$$

-20 olish uchun 16 dan 36 ni ayirish.

$$2x^{2}-12x=-20$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-12x}{2}=-\frac{20}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{12}{2}\right)x=-\frac{20}{2}$$

-12 ni 2 ga bo'lish.

$$x^{2}-6x=-\frac{20}{2}$$

-20 ni 2 ga bo'lish.

$$x^{2}-6x=-10$$

-6 ni bo‘lish, x shartining koeffitsienti, 2 ga -3 olish uchun. Keyin, -3 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-6x+\left(-3\right)^{2}=-10+\left(-3\right)^{2}$$

-3 kvadratini chiqarish.

$$x^{2}-6x+9=-10+9$$

-10 ni 9 ga qo'shish.

$$x^{2}-6x+9=-1$$

x^{2}-6x+9 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-3\right)^{2}=-1$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-3=i x-3=-i$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$x=3+i x=3-i$$