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

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

x uchun yechish

$$x=\frac{1}{4}=0,25$$

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.

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

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

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

-2 kvadratini chiqarish.

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

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-16\times \frac{1}{4}}}{2\times 4}$$

-16 ni \frac{1}{4} marotabaga ko'paytirish.

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

4 ni -4 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{0}}{2\times 4}$$

0 ning kvadrat ildizini chiqarish.

$$x=-\frac{-2}{2\times 4}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{2}{8}$$

\frac{2}{8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{1}{4}$$

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.

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

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

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

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

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

Ikki tarafini 4 ga bo‘ling.

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

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

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

\frac{-2}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

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

$$x^{2}-\frac{1}{2}x=-\frac{1}{16}$$

-\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{4} olish uchun. Keyin, -\frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=-\frac{1}{16}+\left(-\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{4} kvadratini chiqarish.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{-1+1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{16} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=0$$

x^{2}-\frac{1}{2}x+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{4}\right)^{2}=0$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{0}$$

Qisqartirish.

$$x-\frac{1}{4}=0 x-\frac{1}{4}=0$$

\frac{1}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1}{4} x=\frac{1}{4}$$

Tenglama yechildi. Yechimlar bir xil.

$$x=\frac{1}{4}$$