4x-2x^2=7 eching

$$4 x - 2 x ^ { 2 } = 7$$

x uchun yechish (complex solution)

$$x=-\frac{\sqrt{10}i}{2}+1\approx 1-1,58113883i$$

$$x=\frac{\sqrt{10}i}{2}+1\approx 1+1,58113883i$$

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.

$$-2x^{2}+4x=7$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$-2x^{2}+4x-7=7-7$$

O‘zidan 7 ayirilsa 0 qoladi.

$$-2x^{2}+4x-7=0$$

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

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

4 kvadratini chiqarish.

$$x=\frac{-4±\sqrt{16-4\left(-2\right)\left(-7\right)}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16+8\left(-7\right)}}{2\left(-2\right)}$$

8 ni -7 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16-56}}{2\left(-2\right)}$$

16 ni -56 ga qo'shish.

$$x=\frac{-4±\sqrt{-40}}{2\left(-2\right)}$$

-40 ning kvadrat ildizini chiqarish.

$$x=\frac{-4±2\sqrt{10}i}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-4±2\sqrt{10}i}{-4}$$

x=\frac{-4±2\sqrt{10}i}{-4} tenglamasini yeching, bunda ± musbat. -4 ni 2i\sqrt{10} ga qo'shish.

$$x=\frac{-4+2\sqrt{10}i}{-4}$$

-4+2i\sqrt{10} ni -4 ga bo'lish.

$$x=-\frac{\sqrt{10}i}{2}+1$$

x=\frac{-4±2\sqrt{10}i}{-4} tenglamasini yeching, bunda ± manfiy. -4 dan 2i\sqrt{10} ni ayirish.

$$x=\frac{-2\sqrt{10}i-4}{-4}$$

-4-2i\sqrt{10} ni -4 ga bo'lish.

$$x=\frac{\sqrt{10}i}{2}+1$$

Tenglama yechildi.

$$x=-\frac{\sqrt{10}i}{2}+1 x=\frac{\sqrt{10}i}{2}+1$$

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.

$$-2x^{2}+4x=7$$

Ikki tarafini -2 ga bo‘ling.

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

-2 ga bo'lish -2 ga ko'paytirishni bekor qiladi.

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

4 ni -2 ga bo'lish.

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

7 ni -2 ga bo'lish.

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

-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.

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

-\frac{7}{2} ni 1 ga qo'shish.

$$x^{2}-2x+1=-\frac{5}{2}$$

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.

$$\left(x-1\right)^{2}=-\frac{5}{2}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-\frac{5}{2}}$$

Qisqartirish.

$$x-1=\frac{\sqrt{10}i}{2} x-1=-\frac{\sqrt{10}i}{2}$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{10}i}{2}+1 x=-\frac{\sqrt{10}i}{2}+1$$