x^2-4x+16=0 eching

$${ x }^{ 2 } -4x+16=0$$

x uchun yechish (complex solution)

$$x=2+2\sqrt{3}i\approx 2+3,464101615i$$

$$x=-2\sqrt{3}i+2\approx 2-3,464101615i$$

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.

$$x^{2}-4x+16=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -4 ni b va 16 ni c bilan almashtiring.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 16}}{2}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 16}}{2}$$

-4 ni 16 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-64}}{2}$$

16 ni -64 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{-48}}{2}$$

-48 ning kvadrat ildizini chiqarish.

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

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±4\sqrt{3}i}{2}$$

x=\frac{4±4\sqrt{3}i}{2} tenglamasini yeching, bunda ± musbat. 4 ni 4i\sqrt{3} ga qo'shish.

$$x=\frac{4+4\sqrt{3}i}{2}$$

4+4i\sqrt{3} ni 2 ga bo'lish.

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

x=\frac{4±4\sqrt{3}i}{2} tenglamasini yeching, bunda ± manfiy. 4 dan 4i\sqrt{3} ni ayirish.

$$x=\frac{-4\sqrt{3}i+4}{2}$$

4-4i\sqrt{3} ni 2 ga bo'lish.

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

Tenglama yechildi.

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

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.

$$x^{2}-4x+16=0$$

Tenglamaning ikkala tarafidan 16 ni ayirish.

$$x^{2}-4x+16-16=-16$$

O‘zidan 16 ayirilsa 0 qoladi.

$$x^{2}-4x=-16$$

-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.

$$x^{2}-4x+\left(-2\right)^{2}=-16+\left(-2\right)^{2}$$

-2 kvadratini chiqarish.

$$x^{2}-4x+4=-16+4$$

-16 ni 4 ga qo'shish.

$$x^{2}-4x+4=-12$$

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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-12}$$

Qisqartirish.

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

2 ni tenglamaning ikkala tarafiga qo'shish.

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