(x-3)^2+(x-1)^2=0 eching

$$( x - 3 ) ^ { 2 } + ( x - 1 ) ^ { 2 } = 0$$

x uchun yechish (complex solution)

$$x=2+i$$

$$x=2-i$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-3\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-6x+9+\left(x-1\right)^{2}=0$$

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

$$x^{2}-6x+9+x^{2}-2x+1=0$$

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

$$2x^{2}-6x+9-2x+1=0$$

-8x ni olish uchun -6x va -2x ni birlashtirish.

$$2x^{2}-8x+9+1=0$$

10 olish uchun 9 va 1'ni qo'shing.

$$2x^{2}-8x+10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -8 ni b va 10 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\times 10}}{2\times 2}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 2\times 10}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-8\times 10}}{2\times 2}$$

-8 ni 10 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-80}}{2\times 2}$$

64 ni -80 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{-16}}{2\times 2}$$

-16 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±4i}{2\times 2}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±4i}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{8±4i}{4}$$

x=\frac{8±4i}{4} tenglamasini yeching, bunda ± musbat. 8 ni 4i ga qo'shish.

$$x=\frac{8+4i}{4}$$

8+4i ni 4 ga bo'lish.

$$x=2+i$$

x=\frac{8±4i}{4} tenglamasini yeching, bunda ± manfiy. 8 dan 4i ni ayirish.

$$x=\frac{8-4i}{4}$$

8-4i ni 4 ga bo'lish.

$$x=2-i$$

Tenglama yechildi.

$$x=2+i x=2-i$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-3\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}-6x+9+\left(x-1\right)^{2}=0$$

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

$$x^{2}-6x+9+x^{2}-2x+1=0$$

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

$$2x^{2}-6x+9-2x+1=0$$

-8x ni olish uchun -6x va -2x ni birlashtirish.

$$2x^{2}-8x+9+1=0$$

10 olish uchun 9 va 1'ni qo'shing.

$$2x^{2}-8x+10=0$$

Ikkala tarafdan 10 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$2x^{2}-8x=-10$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-8x}{2}=-\frac{10}{2}$$

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

$$x^{2}+\left(-\frac{8}{2}\right)x=-\frac{10}{2}$$

-8 ni 2 ga bo'lish.

$$x^{2}-4x=-\frac{10}{2}$$

-10 ni 2 ga bo'lish.

$$x^{2}-4x=-5$$

-4 ni bo‘lish, x shartining koeffitsienti, 2 ga -2 olish uchun. Keyin, -2 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-4x+\left(-2\right)^{2}=-5+\left(-2\right)^{2}$$

-2 kvadratini chiqarish.

$$x^{2}-4x+4=-5+4$$

-5 ni 4 ga qo'shish.

$$x^{2}-4x+4=-1$$

x^{2}-4x+4 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-2=i x-2=-i$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=2+i x=2-i$$