7x^2+2x+9=8 eching

$$7 { x }^{ 2 } +2x+9=8$$

x uchun yechish (complex solution)

$$x=\frac{-1+\sqrt{6}i}{7}\approx -0,142857143+0,349927106i$$

$$x=\frac{-\sqrt{6}i-1}{7}\approx -0,142857143-0,349927106i$$

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.

$$7x^{2}+2x+9=8$$

Tenglamaning ikkala tarafidan 8 ni ayirish.

$$7x^{2}+2x+9-8=8-8$$

O‘zidan 8 ayirilsa 0 qoladi.

$$7x^{2}+2x+9-8=0$$

9 dan 8 ni ayirish.

$$7x^{2}+2x+1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, 2 ni b va 1 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\times 7}}{2\times 7}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\times 7}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-28}}{2\times 7}$$

4 ni -28 ga qo'shish.

$$x=\frac{-2±\sqrt{-24}}{2\times 7}$$

-24 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{6}i}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{-2±2\sqrt{6}i}{14}$$

x=\frac{-2±2\sqrt{6}i}{14} tenglamasini yeching, bunda ± musbat. -2 ni 2i\sqrt{6} ga qo'shish.

$$x=\frac{-2+2\sqrt{6}i}{14}$$

-2+2i\sqrt{6} ni 14 ga bo'lish.

$$x=\frac{-1+\sqrt{6}i}{7}$$

x=\frac{-2±2\sqrt{6}i}{14} tenglamasini yeching, bunda ± manfiy. -2 dan 2i\sqrt{6} ni ayirish.

$$x=\frac{-2\sqrt{6}i-2}{14}$$

-2-2i\sqrt{6} ni 14 ga bo'lish.

$$x=\frac{-\sqrt{6}i-1}{7}$$

Tenglama yechildi.

$$x=\frac{-1+\sqrt{6}i}{7} x=\frac{-\sqrt{6}i-1}{7}$$

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.

$$7x^{2}+2x+9=8$$

Tenglamaning ikkala tarafidan 9 ni ayirish.

$$7x^{2}+2x+9-9=8-9$$

O‘zidan 9 ayirilsa 0 qoladi.

$$7x^{2}+2x=8-9$$

8 dan 9 ni ayirish.

$$7x^{2}+2x=-1$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7x^{2}+2x}{7}=-\frac{1}{7}$$

7 ga bo'lish 7 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{2}{7}x=-\frac{1}{7}$$

\frac{2}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{7} olish uchun. Keyin, \frac{1}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{2}{7}x+\left(\frac{1}{7}\right)^{2}=-\frac{1}{7}+\left(\frac{1}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{7} kvadratini chiqarish.

$$x^{2}+\frac{2}{7}x+\frac{1}{49}=-\frac{1}{7}+\frac{1}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{7} ni \frac{1}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{2}{7}x+\frac{1}{49}=-\frac{6}{49}$$

x^{2}+\frac{2}{7}x+\frac{1}{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.

$$\left(x+\frac{1}{7}\right)^{2}=-\frac{6}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{7}\right)^{2}}=\sqrt{-\frac{6}{49}}$$

Qisqartirish.

$$x+\frac{1}{7}=\frac{\sqrt{6}i}{7} x+\frac{1}{7}=-\frac{\sqrt{6}i}{7}$$

Tenglamaning ikkala tarafidan \frac{1}{7} ni ayirish.

$$x=\frac{-1+\sqrt{6}i}{7} x=\frac{-\sqrt{6}i-1}{7}$$