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

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

x uchun yechish

$$x=\sqrt{2}+2\approx 3,414213562$$

$$x=2-\sqrt{2}\approx 0,585786438$$

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.

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

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

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

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

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

1 ni -\frac{1}{2} ga qo'shish.

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

\frac{1}{2} ning kvadrat ildizini chiqarish.

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

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\frac{\sqrt{2}}{2}}{2\times \frac{1}{4}}$$

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

$$x=\frac{1±\frac{\sqrt{2}}{2}}{\frac{1}{2}}$$

x=\frac{1±\frac{\sqrt{2}}{2}}{\frac{1}{2}} tenglamasini yeching, bunda ± musbat. 1 ni \frac{\sqrt{2}}{2} ga qo'shish.

$$x=\frac{\frac{\sqrt{2}}{2}+1}{\frac{1}{2}}$$

1+\frac{\sqrt{2}}{2} ni \frac{1}{2} ga bo'lish 1+\frac{\sqrt{2}}{2} ga k'paytirish \frac{1}{2} ga qaytarish.

$$x=\sqrt{2}+2$$

x=\frac{1±\frac{\sqrt{2}}{2}}{\frac{1}{2}} tenglamasini yeching, bunda ± manfiy. 1 dan \frac{\sqrt{2}}{2} ni ayirish.

$$x=\frac{-\frac{\sqrt{2}}{2}+1}{\frac{1}{2}}$$

1-\frac{\sqrt{2}}{2} ni \frac{1}{2} ga bo'lish 1-\frac{\sqrt{2}}{2} ga k'paytirish \frac{1}{2} ga qaytarish.

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

Tenglama yechildi.

$$x=\sqrt{2}+2 x=2-\sqrt{2}$$

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.

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

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

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

O‘zidan \frac{1}{2} ayirilsa 0 qoladi.

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

Ikkala tarafini 4 ga ko‘paytiring.

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

\frac{1}{4} ga bo'lish \frac{1}{4} ga ko'paytirishni bekor qiladi.

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

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

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

-\frac{1}{2} ni \frac{1}{4} ga bo'lish -\frac{1}{2} ga k'paytirish \frac{1}{4} ga qaytarish.

$$x^{2}-4x=-2$$

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

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

-2 kvadratini chiqarish.

$$x^{2}-4x+4=-2+4$$

-2 ni 4 ga qo'shish.

$$x^{2}-4x+4=2$$

x^{2}-4x+4 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

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

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\sqrt{2}+2 x=2-\sqrt{2}$$