7x^2-2x-3=0 eching

$$7 x ^ { 2 } - 2 x - 3 = 0$$

x uchun yechish

$$x=\frac{\sqrt{22}+1}{7}\approx 0,812916537$$

$$x=\frac{1-\sqrt{22}}{7}\approx -0,527202251$$

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.

$$7x^{2}-2x-3=0$$

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 -3 ni c bilan almashtiring.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 7\left(-3\right)}}{2\times 7}$$

-2 kvadratini chiqarish.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 7\left(-3\right)}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-28\left(-3\right)}}{2\times 7}$$

-28 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4+84}}{2\times 7}$$

4 ni 84 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{88}}{2\times 7}$$

88 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±2\sqrt{22}}{2\times 7}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±2\sqrt{22}}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{2±2\sqrt{22}}{14}$$

x=\frac{2±2\sqrt{22}}{14} tenglamasini yeching, bunda ± musbat. 2 ni 2\sqrt{22} ga qo'shish.

$$x=\frac{2\sqrt{22}+2}{14}$$

2+2\sqrt{22} ni 14 ga bo'lish.

$$x=\frac{\sqrt{22}+1}{7}$$

x=\frac{2±2\sqrt{22}}{14} tenglamasini yeching, bunda ± manfiy. 2 dan 2\sqrt{22} ni ayirish.

$$x=\frac{2-2\sqrt{22}}{14}$$

2-2\sqrt{22} ni 14 ga bo'lish.

$$x=\frac{1-\sqrt{22}}{7}$$

Tenglama yechildi.

$$x=\frac{\sqrt{22}+1}{7} x=\frac{1-\sqrt{22}}{7}$$

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.

$$7x^{2}-2x-3=0$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$7x^{2}-2x-3-\left(-3\right)=-\left(-3\right)$$

O‘zidan -3 ayirilsa 0 qoladi.

$$7x^{2}-2x=-\left(-3\right)$$

0 dan -3 ni ayirish.

$$7x^{2}-2x=3$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7x^{2}-2x}{7}=\frac{3}{7}$$

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

$$x^{2}-\frac{2}{7}x=\frac{3}{7}$$

-\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.

$$x^{2}-\frac{2}{7}x+\left(-\frac{1}{7}\right)^{2}=\frac{3}{7}+\left(-\frac{1}{7}\right)^{2}$$

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

$$x^{2}-\frac{2}{7}x+\frac{1}{49}=\frac{3}{7}+\frac{1}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{7} ni \frac{1}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{2}{7}x+\frac{1}{49}=\frac{22}{49}$$

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.

$$\left(x-\frac{1}{7}\right)^{2}=\frac{22}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{7}\right)^{2}}=\sqrt{\frac{22}{49}}$$

Qisqartirish.

$$x-\frac{1}{7}=\frac{\sqrt{22}}{7} x-\frac{1}{7}=-\frac{\sqrt{22}}{7}$$

\frac{1}{7} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{22}+1}{7} x=\frac{1-\sqrt{22}}{7}$$