x^2+12x+23=0 eching

$$x ^ { 2 } + 12 x + 23 = 0$$

x uchun yechish (complex solution)

$$x=\sqrt{13}-6\approx -2,394448725$$

$$x=-\left(\sqrt{13}+6\right)\approx -9,605551275$$

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.

$$x^{2}+12x+23=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 12 ni b va 23 ni c bilan almashtiring.

$$x=\frac{-12±\sqrt{12^{2}-4\times 23}}{2}$$

12 kvadratini chiqarish.

$$x=\frac{-12±\sqrt{144-4\times 23}}{2}$$

-4 ni 23 marotabaga ko'paytirish.

$$x=\frac{-12±\sqrt{144-92}}{2}$$

144 ni -92 ga qo'shish.

$$x=\frac{-12±\sqrt{52}}{2}$$

52 ning kvadrat ildizini chiqarish.

$$x=\frac{-12±2\sqrt{13}}{2}$$

x=\frac{-12±2\sqrt{13}}{2} tenglamasini yeching, bunda ± musbat. -12 ni 2\sqrt{13} ga qo'shish.

$$x=\frac{2\sqrt{13}-12}{2}$$

-12+2\sqrt{13} ni 2 ga bo'lish.

$$x=\sqrt{13}-6$$

x=\frac{-12±2\sqrt{13}}{2} tenglamasini yeching, bunda ± manfiy. -12 dan 2\sqrt{13} ni ayirish.

$$x=\frac{-2\sqrt{13}-12}{2}$$

-12-2\sqrt{13} ni 2 ga bo'lish.

$$x=-\sqrt{13}-6$$

Tenglama yechildi.

$$x=\sqrt{13}-6 x=-\sqrt{13}-6$$

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.

$$x^{2}+12x+23=0$$

Tenglamaning ikkala tarafidan 23 ni ayirish.

$$x^{2}+12x+23-23=-23$$

O‘zidan 23 ayirilsa 0 qoladi.

$$x^{2}+12x=-23$$

12 ni bo‘lish, x shartining koeffitsienti, 2 ga 6 olish uchun. Keyin, 6 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+12x+6^{2}=-23+6^{2}$$

6 kvadratini chiqarish.

$$x^{2}+12x+36=-23+36$$

-23 ni 36 ga qo'shish.

$$x^{2}+12x+36=13$$

x^{2}+12x+36 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+6\right)^{2}=13$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{13}$$

Qisqartirish.

$$x+6=\sqrt{13} x+6=-\sqrt{13}$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$x=\sqrt{13}-6 x=-\sqrt{13}-6$$

x uchun yechish

$$x=\sqrt{13}-6\approx -2,394448725$$

$$x=-\sqrt{13}-6\approx -9,605551275$$

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.

$$x^{2}+12x+23=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 12 ni b va 23 ni c bilan almashtiring.

$$x=\frac{-12±\sqrt{12^{2}-4\times 23}}{2}$$

12 kvadratini chiqarish.

$$x=\frac{-12±\sqrt{144-4\times 23}}{2}$$

-4 ni 23 marotabaga ko'paytirish.

$$x=\frac{-12±\sqrt{144-92}}{2}$$

144 ni -92 ga qo'shish.

$$x=\frac{-12±\sqrt{52}}{2}$$

52 ning kvadrat ildizini chiqarish.

$$x=\frac{-12±2\sqrt{13}}{2}$$

x=\frac{-12±2\sqrt{13}}{2} tenglamasini yeching, bunda ± musbat. -12 ni 2\sqrt{13} ga qo'shish.

$$x=\frac{2\sqrt{13}-12}{2}$$

-12+2\sqrt{13} ni 2 ga bo'lish.

$$x=\sqrt{13}-6$$

x=\frac{-12±2\sqrt{13}}{2} tenglamasini yeching, bunda ± manfiy. -12 dan 2\sqrt{13} ni ayirish.

$$x=\frac{-2\sqrt{13}-12}{2}$$

-12-2\sqrt{13} ni 2 ga bo'lish.

$$x=-\sqrt{13}-6$$

Tenglama yechildi.

$$x=\sqrt{13}-6 x=-\sqrt{13}-6$$

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.

$$x^{2}+12x+23=0$$

Tenglamaning ikkala tarafidan 23 ni ayirish.

$$x^{2}+12x+23-23=-23$$

O‘zidan 23 ayirilsa 0 qoladi.

$$x^{2}+12x=-23$$

12 ni bo‘lish, x shartining koeffitsienti, 2 ga 6 olish uchun. Keyin, 6 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+12x+6^{2}=-23+6^{2}$$

6 kvadratini chiqarish.

$$x^{2}+12x+36=-23+36$$

-23 ni 36 ga qo'shish.

$$x^{2}+12x+36=13$$

x^{2}+12x+36 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+6\right)^{2}=13$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{13}$$

Qisqartirish.

$$x+6=\sqrt{13} x+6=-\sqrt{13}$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$x=\sqrt{13}-6 x=-\sqrt{13}-6$$