x^2+2x+3=12 eching

$${ x }^{ 2 } +2x+3=12$$

x uchun yechish (complex solution)

$$x=\sqrt{10}-1\approx 2,16227766$$

$$x=-\left(\sqrt{10}+1\right)\approx -4,16227766$$

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.

$$x^{2}+2x+3=12$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$x^{2}+2x+3-12=12-12$$

O‘zidan 12 ayirilsa 0 qoladi.

$$x^{2}+2x+3-12=0$$

3 dan 12 ni ayirish.

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

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 -9 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-9\right)}}{2}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\left(-9\right)}}{2}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+36}}{2}$$

4 ni 36 ga qo'shish.

$$x=\frac{-2±\sqrt{40}}{2}$$

40 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{10}}{2}$$

x=\frac{-2±2\sqrt{10}}{2} tenglamasini yeching, bunda ± musbat. -2 ni 2\sqrt{10} ga qo'shish.

$$x=\frac{2\sqrt{10}-2}{2}$$

-2+2\sqrt{10} ni 2 ga bo'lish.

$$x=\sqrt{10}-1$$

x=\frac{-2±2\sqrt{10}}{2} tenglamasini yeching, bunda ± manfiy. -2 dan 2\sqrt{10} ni ayirish.

$$x=\frac{-2\sqrt{10}-2}{2}$$

-2-2\sqrt{10} ni 2 ga bo'lish.

$$x=-\sqrt{10}-1$$

Tenglama yechildi.

$$x=\sqrt{10}-1 x=-\sqrt{10}-1$$

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.

$$x^{2}+2x+3=12$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$x^{2}+2x+3-3=12-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$x^{2}+2x=12-3$$

12 dan 3 ni ayirish.

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

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.

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

1 kvadratini chiqarish.

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

9 ni 1 ga qo'shish.

$$x^{2}+2x+1=10$$

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.

$$\left(x+1\right)^{2}=10$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{10}$$

Qisqartirish.

$$x+1=\sqrt{10} x+1=-\sqrt{10}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=\sqrt{10}-1 x=-\sqrt{10}-1$$

x uchun yechish

$$x=\sqrt{10}-1\approx 2,16227766$$

$$x=-\sqrt{10}-1\approx -4,16227766$$

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.

$$x^{2}+2x+3=12$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$x^{2}+2x+3-12=12-12$$

O‘zidan 12 ayirilsa 0 qoladi.

$$x^{2}+2x+3-12=0$$

3 dan 12 ni ayirish.

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

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 -9 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-9\right)}}{2}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\left(-9\right)}}{2}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+36}}{2}$$

4 ni 36 ga qo'shish.

$$x=\frac{-2±\sqrt{40}}{2}$$

40 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{10}}{2}$$

x=\frac{-2±2\sqrt{10}}{2} tenglamasini yeching, bunda ± musbat. -2 ni 2\sqrt{10} ga qo'shish.

$$x=\frac{2\sqrt{10}-2}{2}$$

-2+2\sqrt{10} ni 2 ga bo'lish.

$$x=\sqrt{10}-1$$

x=\frac{-2±2\sqrt{10}}{2} tenglamasini yeching, bunda ± manfiy. -2 dan 2\sqrt{10} ni ayirish.

$$x=\frac{-2\sqrt{10}-2}{2}$$

-2-2\sqrt{10} ni 2 ga bo'lish.

$$x=-\sqrt{10}-1$$

Tenglama yechildi.

$$x=\sqrt{10}-1 x=-\sqrt{10}-1$$

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.

$$x^{2}+2x+3=12$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$x^{2}+2x+3-3=12-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$x^{2}+2x=12-3$$

12 dan 3 ni ayirish.

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

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.

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

1 kvadratini chiqarish.

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

9 ni 1 ga qo'shish.

$$x^{2}+2x+1=10$$

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.

$$\left(x+1\right)^{2}=10$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{10}$$

Qisqartirish.

$$x+1=\sqrt{10} x+1=-\sqrt{10}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=\sqrt{10}-1 x=-\sqrt{10}-1$$