7x^2-14x+1/4=0 eching

$$7 x ^ { 2 } - 14 x + \frac { 1 } { 4 } = 0$$

x uchun yechish

$$x=\frac{3\sqrt{21}}{14}+1\approx 1,981980506$$

$$x=-\frac{3\sqrt{21}}{14}+1\approx 0,018019494$$

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.

$$7x^{2}-14x+\frac{1}{4}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, -14 ni b va \frac{1}{4} ni c bilan almashtiring.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 7\times \frac{1}{4}}}{2\times 7}$$

-14 kvadratini chiqarish.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 7\times \frac{1}{4}}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-28\times \frac{1}{4}}}{2\times 7}$$

-28 ni \frac{1}{4} marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-7}}{2\times 7}$$

196 ni -7 ga qo'shish.

$$x=\frac{-\left(-14\right)±\sqrt{189}}{2\times 7}$$

189 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-14\right)±3\sqrt{21}}{2\times 7}$$

-14 ning teskarisi 14 ga teng.

$$x=\frac{14±3\sqrt{21}}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{14±3\sqrt{21}}{14}$$

x=\frac{14±3\sqrt{21}}{14} tenglamasini yeching, bunda ± musbat. 14 ni 3\sqrt{21} ga qo'shish.

$$x=\frac{3\sqrt{21}+14}{14}$$

14+3\sqrt{21} ni 14 ga bo'lish.

$$x=\frac{3\sqrt{21}}{14}+1$$

x=\frac{14±3\sqrt{21}}{14} tenglamasini yeching, bunda ± manfiy. 14 dan 3\sqrt{21} ni ayirish.

$$x=\frac{14-3\sqrt{21}}{14}$$

14-3\sqrt{21} ni 14 ga bo'lish.

$$x=-\frac{3\sqrt{21}}{14}+1$$

Tenglama yechildi.

$$x=\frac{3\sqrt{21}}{14}+1 x=-\frac{3\sqrt{21}}{14}+1$$

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.

$$7x^{2}-14x+\frac{1}{4}=0$$

Tenglamaning ikkala tarafidan \frac{1}{4} ni ayirish.

$$7x^{2}-14x+\frac{1}{4}-\frac{1}{4}=-\frac{1}{4}$$

O‘zidan \frac{1}{4} ayirilsa 0 qoladi.

$$7x^{2}-14x=-\frac{1}{4}$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7x^{2}-14x}{7}=-\frac{\frac{1}{4}}{7}$$

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

$$x^{2}+\left(-\frac{14}{7}\right)x=-\frac{\frac{1}{4}}{7}$$

-14 ni 7 ga bo'lish.

$$x^{2}-2x=-\frac{\frac{1}{4}}{7}$$

-\frac{1}{4} ni 7 ga bo'lish.

$$x^{2}-2x=-\frac{1}{28}$$

-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.

$$x^{2}-2x+1=-\frac{1}{28}+1$$

-\frac{1}{28} ni 1 ga qo'shish.

$$x^{2}-2x+1=\frac{27}{28}$$

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.

$$\left(x-1\right)^{2}=\frac{27}{28}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{27}{28}}$$

Qisqartirish.

$$x-1=\frac{3\sqrt{21}}{14} x-1=-\frac{3\sqrt{21}}{14}$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3\sqrt{21}}{14}+1 x=-\frac{3\sqrt{21}}{14}+1$$