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

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

x uchun yechish

$$x=\frac{\sqrt{3}}{2}+1\approx 1,866025404$$

$$x=-\frac{\sqrt{3}}{2}+1\approx 0,133974596$$

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.

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

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

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

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

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

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

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

2 kvadratini chiqarish.

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

-4 ni -1 marotabaga ko'paytirish.

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

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

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

4 ni -1 ga qo'shish.

$$x=\frac{-2±\sqrt{3}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{3}}{-2}$$

x=\frac{-2±\sqrt{3}}{-2} tenglamasini yeching, bunda ± musbat. -2 ni \sqrt{3} ga qo'shish.

$$x=\frac{\sqrt{3}-2}{-2}$$

-2+\sqrt{3} ni -2 ga bo'lish.

$$x=-\frac{\sqrt{3}}{2}+1$$

x=\frac{-2±\sqrt{3}}{-2} tenglamasini yeching, bunda ± manfiy. -2 dan \sqrt{3} ni ayirish.

$$x=\frac{-\sqrt{3}-2}{-2}$$

-2-\sqrt{3} ni -2 ga bo'lish.

$$x=\frac{\sqrt{3}}{2}+1$$

Tenglama yechildi.

$$x=-\frac{\sqrt{3}}{2}+1 x=\frac{\sqrt{3}}{2}+1$$

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.

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

Ikki tarafini -1 ga bo‘ling.

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

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

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

2 ni -1 ga bo'lish.

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

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

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

-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.

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

-\frac{1}{4} ni 1 ga qo'shish.

$$x^{2}-2x+1=\frac{3}{4}$$

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.

$$\left(x-1\right)^{2}=\frac{3}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{3}{4}}$$

Qisqartirish.

$$x-1=\frac{\sqrt{3}}{2} x-1=-\frac{\sqrt{3}}{2}$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{3}}{2}+1 x=-\frac{\sqrt{3}}{2}+1$$