x^2+14x-12=0 eching

$$x ^ { 2 } + 14 x - 12 = 0$$

x uchun yechish (complex solution)

$$x=\sqrt{61}-7\approx 0,810249676$$

$$x=-\left(\sqrt{61}+7\right)\approx -14,810249676$$

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.

$$x^{2}+14x-12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 14 ni b va -12 ni c bilan almashtiring.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-12\right)}}{2}$$

14 kvadratini chiqarish.

$$x=\frac{-14±\sqrt{196-4\left(-12\right)}}{2}$$

-4 ni -12 marotabaga ko'paytirish.

$$x=\frac{-14±\sqrt{196+48}}{2}$$

196 ni 48 ga qo'shish.

$$x=\frac{-14±\sqrt{244}}{2}$$

244 ning kvadrat ildizini chiqarish.

$$x=\frac{-14±2\sqrt{61}}{2}$$

x=\frac{-14±2\sqrt{61}}{2} tenglamasini yeching, bunda ± musbat. -14 ni 2\sqrt{61} ga qo'shish.

$$x=\frac{2\sqrt{61}-14}{2}$$

-14+2\sqrt{61} ni 2 ga bo'lish.

$$x=\sqrt{61}-7$$

x=\frac{-14±2\sqrt{61}}{2} tenglamasini yeching, bunda ± manfiy. -14 dan 2\sqrt{61} ni ayirish.

$$x=\frac{-2\sqrt{61}-14}{2}$$

-14-2\sqrt{61} ni 2 ga bo'lish.

$$x=-\sqrt{61}-7$$

Tenglama yechildi.

$$x=\sqrt{61}-7 x=-\sqrt{61}-7$$

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.

$$x^{2}+14x-12=0$$

12 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}+14x-12-\left(-12\right)=-\left(-12\right)$$

O‘zidan -12 ayirilsa 0 qoladi.

$$x^{2}+14x=-\left(-12\right)$$

0 dan -12 ni ayirish.

$$x^{2}+14x=12$$

14 ni bo‘lish, x shartining koeffitsienti, 2 ga 7 olish uchun. Keyin, 7 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+14x+7^{2}=12+7^{2}$$

7 kvadratini chiqarish.

$$x^{2}+14x+49=12+49$$

12 ni 49 ga qo'shish.

$$x^{2}+14x+49=61$$

x^{2}+14x+49 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+7\right)^{2}=61$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{61}$$

Qisqartirish.

$$x+7=\sqrt{61} x+7=-\sqrt{61}$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$x=\sqrt{61}-7 x=-\sqrt{61}-7$$

x uchun yechish

$$x=\sqrt{61}-7\approx 0,810249676$$

$$x=-\sqrt{61}-7\approx -14,810249676$$

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.

$$x^{2}+14x-12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 14 ni b va -12 ni c bilan almashtiring.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-12\right)}}{2}$$

14 kvadratini chiqarish.

$$x=\frac{-14±\sqrt{196-4\left(-12\right)}}{2}$$

-4 ni -12 marotabaga ko'paytirish.

$$x=\frac{-14±\sqrt{196+48}}{2}$$

196 ni 48 ga qo'shish.

$$x=\frac{-14±\sqrt{244}}{2}$$

244 ning kvadrat ildizini chiqarish.

$$x=\frac{-14±2\sqrt{61}}{2}$$

x=\frac{-14±2\sqrt{61}}{2} tenglamasini yeching, bunda ± musbat. -14 ni 2\sqrt{61} ga qo'shish.

$$x=\frac{2\sqrt{61}-14}{2}$$

-14+2\sqrt{61} ni 2 ga bo'lish.

$$x=\sqrt{61}-7$$

x=\frac{-14±2\sqrt{61}}{2} tenglamasini yeching, bunda ± manfiy. -14 dan 2\sqrt{61} ni ayirish.

$$x=\frac{-2\sqrt{61}-14}{2}$$

-14-2\sqrt{61} ni 2 ga bo'lish.

$$x=-\sqrt{61}-7$$

Tenglama yechildi.

$$x=\sqrt{61}-7 x=-\sqrt{61}-7$$

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.

$$x^{2}+14x-12=0$$

12 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}+14x-12-\left(-12\right)=-\left(-12\right)$$

O‘zidan -12 ayirilsa 0 qoladi.

$$x^{2}+14x=-\left(-12\right)$$

0 dan -12 ni ayirish.

$$x^{2}+14x=12$$

14 ni bo‘lish, x shartining koeffitsienti, 2 ga 7 olish uchun. Keyin, 7 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+14x+7^{2}=12+7^{2}$$

7 kvadratini chiqarish.

$$x^{2}+14x+49=12+49$$

12 ni 49 ga qo'shish.

$$x^{2}+14x+49=61$$

x^{2}+14x+49 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+7\right)^{2}=61$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{61}$$

Qisqartirish.

$$x+7=\sqrt{61} x+7=-\sqrt{61}$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$x=\sqrt{61}-7 x=-\sqrt{61}-7$$