x^2+2x+11=111 eching

$$x ^ { 2 } + 2 x + 11 = 111$$

x uchun yechish (complex solution)

$$x=\sqrt{101}-1\approx 9,049875621$$

$$x=-\left(\sqrt{101}+1\right)\approx -11,049875621$$

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.

$$x^{2}+2x+11=111$$

Tenglamaning ikkala tarafidan 111 ni ayirish.

$$x^{2}+2x+11-111=111-111$$

O‘zidan 111 ayirilsa 0 qoladi.

$$x^{2}+2x+11-111=0$$

11 dan 111 ni ayirish.

$$x^{2}+2x-100=0$$

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 -100 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-100\right)}}{2}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\left(-100\right)}}{2}$$

-4 ni -100 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+400}}{2}$$

4 ni 400 ga qo'shish.

$$x=\frac{-2±\sqrt{404}}{2}$$

404 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{101}}{2}$$

x=\frac{-2±2\sqrt{101}}{2} tenglamasini yeching, bunda ± musbat. -2 ni 2\sqrt{101} ga qo'shish.

$$x=\frac{2\sqrt{101}-2}{2}$$

-2+2\sqrt{101} ni 2 ga bo'lish.

$$x=\sqrt{101}-1$$

x=\frac{-2±2\sqrt{101}}{2} tenglamasini yeching, bunda ± manfiy. -2 dan 2\sqrt{101} ni ayirish.

$$x=\frac{-2\sqrt{101}-2}{2}$$

-2-2\sqrt{101} ni 2 ga bo'lish.

$$x=-\sqrt{101}-1$$

Tenglama yechildi.

$$x=\sqrt{101}-1 x=-\sqrt{101}-1$$

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.

$$x^{2}+2x+11=111$$

Tenglamaning ikkala tarafidan 11 ni ayirish.

$$x^{2}+2x+11-11=111-11$$

O‘zidan 11 ayirilsa 0 qoladi.

$$x^{2}+2x=111-11$$

111 dan 11 ni ayirish.

$$x^{2}+2x=100$$

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.

$$x^{2}+2x+1^{2}=100+1^{2}$$

1 kvadratini chiqarish.

$$x^{2}+2x+1=100+1$$

100 ni 1 ga qo'shish.

$$x^{2}+2x+1=101$$

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.

$$\left(x+1\right)^{2}=101$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{101}$$

Qisqartirish.

$$x+1=\sqrt{101} x+1=-\sqrt{101}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=\sqrt{101}-1 x=-\sqrt{101}-1$$

x uchun yechish

$$x=\sqrt{101}-1\approx 9,049875621$$

$$x=-\sqrt{101}-1\approx -11,049875621$$

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.

$$x^{2}+2x+11=111$$

Tenglamaning ikkala tarafidan 111 ni ayirish.

$$x^{2}+2x+11-111=111-111$$

O‘zidan 111 ayirilsa 0 qoladi.

$$x^{2}+2x+11-111=0$$

11 dan 111 ni ayirish.

$$x^{2}+2x-100=0$$

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 -100 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-100\right)}}{2}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\left(-100\right)}}{2}$$

-4 ni -100 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+400}}{2}$$

4 ni 400 ga qo'shish.

$$x=\frac{-2±\sqrt{404}}{2}$$

404 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{101}}{2}$$

x=\frac{-2±2\sqrt{101}}{2} tenglamasini yeching, bunda ± musbat. -2 ni 2\sqrt{101} ga qo'shish.

$$x=\frac{2\sqrt{101}-2}{2}$$

-2+2\sqrt{101} ni 2 ga bo'lish.

$$x=\sqrt{101}-1$$

x=\frac{-2±2\sqrt{101}}{2} tenglamasini yeching, bunda ± manfiy. -2 dan 2\sqrt{101} ni ayirish.

$$x=\frac{-2\sqrt{101}-2}{2}$$

-2-2\sqrt{101} ni 2 ga bo'lish.

$$x=-\sqrt{101}-1$$

Tenglama yechildi.

$$x=\sqrt{101}-1 x=-\sqrt{101}-1$$

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.

$$x^{2}+2x+11=111$$

Tenglamaning ikkala tarafidan 11 ni ayirish.

$$x^{2}+2x+11-11=111-11$$

O‘zidan 11 ayirilsa 0 qoladi.

$$x^{2}+2x=111-11$$

111 dan 11 ni ayirish.

$$x^{2}+2x=100$$

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.

$$x^{2}+2x+1^{2}=100+1^{2}$$

1 kvadratini chiqarish.

$$x^{2}+2x+1=100+1$$

100 ni 1 ga qo'shish.

$$x^{2}+2x+1=101$$

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.

$$\left(x+1\right)^{2}=101$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{101}$$

Qisqartirish.

$$x+1=\sqrt{101} x+1=-\sqrt{101}$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=\sqrt{101}-1 x=-\sqrt{101}-1$$