x^2+18x+67=0 eching

$$x ^ { 2 } + 18 x + 67 = 0$$

x uchun yechish (complex solution)

$$x=\sqrt{14}-9\approx -5,258342613$$

$$x=-\left(\sqrt{14}+9\right)\approx -12,741657387$$

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.

$$x^{2}+18x+67=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 18 ni b va 67 ni c bilan almashtiring.

$$x=\frac{-18±\sqrt{18^{2}-4\times 67}}{2}$$

18 kvadratini chiqarish.

$$x=\frac{-18±\sqrt{324-4\times 67}}{2}$$

-4 ni 67 marotabaga ko'paytirish.

$$x=\frac{-18±\sqrt{324-268}}{2}$$

324 ni -268 ga qo'shish.

$$x=\frac{-18±\sqrt{56}}{2}$$

56 ning kvadrat ildizini chiqarish.

$$x=\frac{-18±2\sqrt{14}}{2}$$

x=\frac{-18±2\sqrt{14}}{2} tenglamasini yeching, bunda ± musbat. -18 ni 2\sqrt{14} ga qo'shish.

$$x=\frac{2\sqrt{14}-18}{2}$$

-18+2\sqrt{14} ni 2 ga bo'lish.

$$x=\sqrt{14}-9$$

x=\frac{-18±2\sqrt{14}}{2} tenglamasini yeching, bunda ± manfiy. -18 dan 2\sqrt{14} ni ayirish.

$$x=\frac{-2\sqrt{14}-18}{2}$$

-18-2\sqrt{14} ni 2 ga bo'lish.

$$x=-\sqrt{14}-9$$

Tenglama yechildi.

$$x=\sqrt{14}-9 x=-\sqrt{14}-9$$

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.

$$x^{2}+18x+67=0$$

Tenglamaning ikkala tarafidan 67 ni ayirish.

$$x^{2}+18x+67-67=-67$$

O‘zidan 67 ayirilsa 0 qoladi.

$$x^{2}+18x=-67$$

18 ni bo‘lish, x shartining koeffitsienti, 2 ga 9 olish uchun. Keyin, 9 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+18x+9^{2}=-67+9^{2}$$

9 kvadratini chiqarish.

$$x^{2}+18x+81=-67+81$$

-67 ni 81 ga qo'shish.

$$x^{2}+18x+81=14$$

x^{2}+18x+81 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+9\right)^{2}=14$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{14}$$

Qisqartirish.

$$x+9=\sqrt{14} x+9=-\sqrt{14}$$

Tenglamaning ikkala tarafidan 9 ni ayirish.

$$x=\sqrt{14}-9 x=-\sqrt{14}-9$$

x uchun yechish

$$x=\sqrt{14}-9\approx -5,258342613$$

$$x=-\sqrt{14}-9\approx -12,741657387$$

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.

$$x^{2}+18x+67=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 18 ni b va 67 ni c bilan almashtiring.

$$x=\frac{-18±\sqrt{18^{2}-4\times 67}}{2}$$

18 kvadratini chiqarish.

$$x=\frac{-18±\sqrt{324-4\times 67}}{2}$$

-4 ni 67 marotabaga ko'paytirish.

$$x=\frac{-18±\sqrt{324-268}}{2}$$

324 ni -268 ga qo'shish.

$$x=\frac{-18±\sqrt{56}}{2}$$

56 ning kvadrat ildizini chiqarish.

$$x=\frac{-18±2\sqrt{14}}{2}$$

x=\frac{-18±2\sqrt{14}}{2} tenglamasini yeching, bunda ± musbat. -18 ni 2\sqrt{14} ga qo'shish.

$$x=\frac{2\sqrt{14}-18}{2}$$

-18+2\sqrt{14} ni 2 ga bo'lish.

$$x=\sqrt{14}-9$$

x=\frac{-18±2\sqrt{14}}{2} tenglamasini yeching, bunda ± manfiy. -18 dan 2\sqrt{14} ni ayirish.

$$x=\frac{-2\sqrt{14}-18}{2}$$

-18-2\sqrt{14} ni 2 ga bo'lish.

$$x=-\sqrt{14}-9$$

Tenglama yechildi.

$$x=\sqrt{14}-9 x=-\sqrt{14}-9$$

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.

$$x^{2}+18x+67=0$$

Tenglamaning ikkala tarafidan 67 ni ayirish.

$$x^{2}+18x+67-67=-67$$

O‘zidan 67 ayirilsa 0 qoladi.

$$x^{2}+18x=-67$$

18 ni bo‘lish, x shartining koeffitsienti, 2 ga 9 olish uchun. Keyin, 9 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+18x+9^{2}=-67+9^{2}$$

9 kvadratini chiqarish.

$$x^{2}+18x+81=-67+81$$

-67 ni 81 ga qo'shish.

$$x^{2}+18x+81=14$$

x^{2}+18x+81 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+9\right)^{2}=14$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{14}$$

Qisqartirish.

$$x+9=\sqrt{14} x+9=-\sqrt{14}$$

Tenglamaning ikkala tarafidan 9 ni ayirish.

$$x=\sqrt{14}-9 x=-\sqrt{14}-9$$