x^2-6x+11=0 eching

$$x ^ { 2 } - 6 x + 11 = 0$$

x uchun yechish (complex solution)

$$x=3+\sqrt{2}i\approx 3+1,414213562i$$

$$x=-\sqrt{2}i+3\approx 3-1,414213562i$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$x^{2}-6x+11=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -6 ni b va 11 ni c bilan almashtiring.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 11}}{2}$$

-6 kvadratini chiqarish.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 11}}{2}$$

-4 ni 11 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36-44}}{2}$$

36 ni -44 ga qo'shish.

$$x=\frac{-\left(-6\right)±\sqrt{-8}}{2}$$

-8 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-6\right)±2\sqrt{2}i}{2}$$

-6 ning teskarisi 6 ga teng.

$$x=\frac{6±2\sqrt{2}i}{2}$$

x=\frac{6±2\sqrt{2}i}{2} tenglamasini yeching, bunda ± musbat. 6 ni 2i\sqrt{2} ga qo'shish.

$$x=\frac{6+2\sqrt{2}i}{2}$$

6+2i\sqrt{2} ni 2 ga bo'lish.

$$x=3+\sqrt{2}i$$

x=\frac{6±2\sqrt{2}i}{2} tenglamasini yeching, bunda ± manfiy. 6 dan 2i\sqrt{2} ni ayirish.

$$x=\frac{-2\sqrt{2}i+6}{2}$$

6-2i\sqrt{2} ni 2 ga bo'lish.

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$x^{2}-6x+11=0$$

Tenglamaning ikkala tarafidan 11 ni ayirish.

$$x^{2}-6x+11-11=-11$$

O‘zidan 11 ayirilsa 0 qoladi.

$$x^{2}-6x=-11$$

-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.

$$x^{2}-6x+\left(-3\right)^{2}=-11+\left(-3\right)^{2}$$

-3 kvadratini chiqarish.

$$x^{2}-6x+9=-11+9$$

-11 ni 9 ga qo'shish.

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

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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-3=\sqrt{2}i x-3=-\sqrt{2}i$$

3 ni tenglamaning ikkala tarafiga qo'shish.

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