x^2+4x-3=12 eching

$$x ^ { 2 } + 4 x - 3 = 12$$

x uchun yechish (complex solution)

$$x=\sqrt{19}-2\approx 2,358898944$$

$$x=-\left(\sqrt{19}+2\right)\approx -6,358898944$$

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.

$$x^{2}+4x-3=12$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$x^{2}+4x-3-12=12-12$$

O‘zidan 12 ayirilsa 0 qoladi.

$$x^{2}+4x-3-12=0$$

-3 dan 12 ni ayirish.

$$x^{2}+4x-15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 4 ni b va -15 ni c bilan almashtiring.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-15\right)}}{2}$$

4 kvadratini chiqarish.

$$x=\frac{-4±\sqrt{16-4\left(-15\right)}}{2}$$

-4 ni -15 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16+60}}{2}$$

16 ni 60 ga qo'shish.

$$x=\frac{-4±\sqrt{76}}{2}$$

76 ning kvadrat ildizini chiqarish.

$$x=\frac{-4±2\sqrt{19}}{2}$$

x=\frac{-4±2\sqrt{19}}{2} tenglamasini yeching, bunda ± musbat. -4 ni 2\sqrt{19} ga qo'shish.

$$x=\frac{2\sqrt{19}-4}{2}$$

-4+2\sqrt{19} ni 2 ga bo'lish.

$$x=\sqrt{19}-2$$

x=\frac{-4±2\sqrt{19}}{2} tenglamasini yeching, bunda ± manfiy. -4 dan 2\sqrt{19} ni ayirish.

$$x=\frac{-2\sqrt{19}-4}{2}$$

-4-2\sqrt{19} ni 2 ga bo'lish.

$$x=-\sqrt{19}-2$$

Tenglama yechildi.

$$x=\sqrt{19}-2 x=-\sqrt{19}-2$$

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.

$$x^{2}+4x-3=12$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}+4x-3-\left(-3\right)=12-\left(-3\right)$$

O‘zidan -3 ayirilsa 0 qoladi.

$$x^{2}+4x=12-\left(-3\right)$$

12 dan -3 ni ayirish.

$$x^{2}+4x=15$$

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

$$x^{2}+4x+2^{2}=15+2^{2}$$

2 kvadratini chiqarish.

$$x^{2}+4x+4=15+4$$

15 ni 4 ga qo'shish.

$$x^{2}+4x+4=19$$

x^{2}+4x+4 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+2\right)^{2}=19$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{19}$$

Qisqartirish.

$$x+2=\sqrt{19} x+2=-\sqrt{19}$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$x=\sqrt{19}-2 x=-\sqrt{19}-2$$

x uchun yechish

$$x=\sqrt{19}-2\approx 2,358898944$$

$$x=-\sqrt{19}-2\approx -6,358898944$$

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.

$$x^{2}+4x-3=12$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$x^{2}+4x-3-12=12-12$$

O‘zidan 12 ayirilsa 0 qoladi.

$$x^{2}+4x-3-12=0$$

-3 dan 12 ni ayirish.

$$x^{2}+4x-15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 4 ni b va -15 ni c bilan almashtiring.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-15\right)}}{2}$$

4 kvadratini chiqarish.

$$x=\frac{-4±\sqrt{16-4\left(-15\right)}}{2}$$

-4 ni -15 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16+60}}{2}$$

16 ni 60 ga qo'shish.

$$x=\frac{-4±\sqrt{76}}{2}$$

76 ning kvadrat ildizini chiqarish.

$$x=\frac{-4±2\sqrt{19}}{2}$$

x=\frac{-4±2\sqrt{19}}{2} tenglamasini yeching, bunda ± musbat. -4 ni 2\sqrt{19} ga qo'shish.

$$x=\frac{2\sqrt{19}-4}{2}$$

-4+2\sqrt{19} ni 2 ga bo'lish.

$$x=\sqrt{19}-2$$

x=\frac{-4±2\sqrt{19}}{2} tenglamasini yeching, bunda ± manfiy. -4 dan 2\sqrt{19} ni ayirish.

$$x=\frac{-2\sqrt{19}-4}{2}$$

-4-2\sqrt{19} ni 2 ga bo'lish.

$$x=-\sqrt{19}-2$$

Tenglama yechildi.

$$x=\sqrt{19}-2 x=-\sqrt{19}-2$$

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.

$$x^{2}+4x-3=12$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}+4x-3-\left(-3\right)=12-\left(-3\right)$$

O‘zidan -3 ayirilsa 0 qoladi.

$$x^{2}+4x=12-\left(-3\right)$$

12 dan -3 ni ayirish.

$$x^{2}+4x=15$$

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

$$x^{2}+4x+2^{2}=15+2^{2}$$

2 kvadratini chiqarish.

$$x^{2}+4x+4=15+4$$

15 ni 4 ga qo'shish.

$$x^{2}+4x+4=19$$

x^{2}+4x+4 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+2\right)^{2}=19$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{19}$$

Qisqartirish.

$$x+2=\sqrt{19} x+2=-\sqrt{19}$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$x=\sqrt{19}-2 x=-\sqrt{19}-2$$