4^2+x^2+2x=9 eching

$${ 4 }^{ 2 } + { x }^{ 2 } +2x=9$$

x uchun yechish (complex solution)

$$x=-1+\sqrt{6}i\approx -1+2,449489743i$$

$$x=-\sqrt{6}i-1\approx -1-2,449489743i$$

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.

$$x^{2}+2x+16=9$$

Tenglamaning ikkala tarafidan 9 ni ayirish.

$$x^{2}+2x+16-9=9-9$$

O‘zidan 9 ayirilsa 0 qoladi.

$$x^{2}+2x+16-9=0$$

16 dan 9 ni ayirish.

$$x^{2}+2x+7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 2 ni b va 7 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\times 7}}{2}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\times 7}}{2}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-28}}{2}$$

4 ni -28 ga qo'shish.

$$x=\frac{-2±\sqrt{-24}}{2}$$

-24 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{6}i}{2}$$

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

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

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

$$x=-1+\sqrt{6}i$$

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

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

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

$$x=-\sqrt{6}i-1$$

Tenglama yechildi.

$$x=-1+\sqrt{6}i x=-\sqrt{6}i-1$$

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.

$$x^{2}+2x+16=9$$

Tenglamaning ikkala tarafidan 16 ni ayirish.

$$x^{2}+2x+16-16=9-16$$

O‘zidan 16 ayirilsa 0 qoladi.

$$x^{2}+2x=9-16$$

9 dan 16 ni ayirish.

$$x^{2}+2x=-7$$

2 ni bo‘lish, x shartining koeffitsienti, 2 ga 1 olish uchun. Keyin, 1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+2x+1^{2}=-7+1^{2}$$

1 kvadratini chiqarish.

$$x^{2}+2x+1=-7+1$$

-7 ni 1 ga qo'shish.

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

x^{2}+2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x+1=\sqrt{6}i x+1=-\sqrt{6}i$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=-1+\sqrt{6}i x=-\sqrt{6}i-1$$