4x=4x^2-8x+4 eching

$$4 x = 4 x ^ { 2 } - 8 x + 4$$

x uchun yechish

$$x = \frac{\sqrt{5} + 3}{2} \approx 2,618033989$$

$$x=\frac{3-\sqrt{5}}{2}\approx 0,381966011$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 4x^{2} ni ayirish.

$$4x-4x^{2}=-8x+4$$

8x ni ikki tarafga qo’shing.

$$4x-4x^{2}+8x=4$$

12x ni olish uchun 4x va 8x ni birlashtirish.

$$12x-4x^{2}=4$$

Ikkala tarafdan 4 ni ayirish.

$$12x-4x^{2}-4=0$$

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.

$$-4x^{2}+12x-4=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -4 ni a, 12 ni b va -4 ni c bilan almashtiring.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-4\right)\left(-4\right)}}{2\left(-4\right)}$$

12 kvadratini chiqarish.

$$x=\frac{-12±\sqrt{144-4\left(-4\right)\left(-4\right)}}{2\left(-4\right)}$$

-4 ni -4 marotabaga ko'paytirish.

$$x=\frac{-12±\sqrt{144+16\left(-4\right)}}{2\left(-4\right)}$$

16 ni -4 marotabaga ko'paytirish.

$$x=\frac{-12±\sqrt{144-64}}{2\left(-4\right)}$$

144 ni -64 ga qo'shish.

$$x=\frac{-12±\sqrt{80}}{2\left(-4\right)}$$

80 ning kvadrat ildizini chiqarish.

$$x=\frac{-12±4\sqrt{5}}{2\left(-4\right)}$$

2 ni -4 marotabaga ko'paytirish.

$$x=\frac{-12±4\sqrt{5}}{-8}$$

x=\frac{-12±4\sqrt{5}}{-8} tenglamasini yeching, bunda ± musbat. -12 ni 4\sqrt{5} ga qo'shish.

$$x=\frac{4\sqrt{5}-12}{-8}$$

-12+4\sqrt{5} ni -8 ga bo'lish.

$$x=\frac{3-\sqrt{5}}{2}$$

x=\frac{-12±4\sqrt{5}}{-8} tenglamasini yeching, bunda ± manfiy. -12 dan 4\sqrt{5} ni ayirish.

$$x=\frac{-4\sqrt{5}-12}{-8}$$

-12-4\sqrt{5} ni -8 ga bo'lish.

$$x=\frac{\sqrt{5}+3}{2}$$

Tenglama yechildi.

$$x=\frac{3-\sqrt{5}}{2} x=\frac{\sqrt{5}+3}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 4x^{2} ni ayirish.

$$4x-4x^{2}=-8x+4$$

8x ni ikki tarafga qo’shing.

$$4x-4x^{2}+8x=4$$

12x ni olish uchun 4x va 8x ni birlashtirish.

$$12x-4x^{2}=4$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-4x^{2}+12x=4$$

Ikki tarafini -4 ga bo‘ling.

$$\frac{-4x^{2}+12x}{-4}=\frac{4}{-4}$$

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

$$x^{2}+\frac{12}{-4}x=\frac{4}{-4}$$

12 ni -4 ga bo'lish.

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

4 ni -4 ga bo'lish.

$$x^{2}-3x=-1$$

-3 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{2} olish uchun. Keyin, -\frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$x^{2}-3x+\frac{9}{4}=-1+\frac{9}{4}$$

-1 ni \frac{9}{4} ga qo'shish.

$$x^{2}-3x+\frac{9}{4}=\frac{5}{4}$$

x^{2}-3x+\frac{9}{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.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{5}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{5}{4}}$$

Qisqartirish.

$$x-\frac{3}{2}=\frac{\sqrt{5}}{2} x-\frac{3}{2}=-\frac{\sqrt{5}}{2}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{5}+3}{2} x=\frac{3-\sqrt{5}}{2}$$