x-2x^2=8 eching

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

x uchun yechish (complex solution)

$$x=\frac{-3\sqrt{7}i+1}{4}\approx 0,25-1,984313483i$$

$$x=\frac{1+3\sqrt{7}i}{4}\approx 0,25+1,984313483i$$

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.

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

Tenglamaning ikkala tarafidan 8 ni ayirish.

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

O‘zidan 8 ayirilsa 0 qoladi.

$$-2x^{2}+x-8=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 1 ni b va -8 ni c bilan almashtiring.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-2\right)\left(-8\right)}}{2\left(-2\right)}$$

1 kvadratini chiqarish.

$$x=\frac{-1±\sqrt{1-4\left(-2\right)\left(-8\right)}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1+8\left(-8\right)}}{2\left(-2\right)}$$

8 ni -8 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1-64}}{2\left(-2\right)}$$

1 ni -64 ga qo'shish.

$$x=\frac{-1±\sqrt{-63}}{2\left(-2\right)}$$

-63 ning kvadrat ildizini chiqarish.

$$x=\frac{-1±3\sqrt{7}i}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-1±3\sqrt{7}i}{-4}$$

x=\frac{-1±3\sqrt{7}i}{-4} tenglamasini yeching, bunda ± musbat. -1 ni 3i\sqrt{7} ga qo'shish.

$$x=\frac{-1+3\sqrt{7}i}{-4}$$

-1+3i\sqrt{7} ni -4 ga bo'lish.

$$x=\frac{-3\sqrt{7}i+1}{4}$$

x=\frac{-1±3\sqrt{7}i}{-4} tenglamasini yeching, bunda ± manfiy. -1 dan 3i\sqrt{7} ni ayirish.

$$x=\frac{-3\sqrt{7}i-1}{-4}$$

-1-3i\sqrt{7} ni -4 ga bo'lish.

$$x=\frac{1+3\sqrt{7}i}{4}$$

Tenglama yechildi.

$$x=\frac{-3\sqrt{7}i+1}{4} x=\frac{1+3\sqrt{7}i}{4}$$

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.

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

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}+x}{-2}=\frac{8}{-2}$$

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

$$x^{2}+\frac{1}{-2}x=\frac{8}{-2}$$

1 ni -2 ga bo'lish.

$$x^{2}-\frac{1}{2}x=\frac{8}{-2}$$

8 ni -2 ga bo'lish.

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

-\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.

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

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

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-4+\frac{1}{16}$$

-4 ni \frac{1}{16} ga qo'shish.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{63}{16}$$

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.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{63}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{63}{16}}$$

Qisqartirish.

$$x-\frac{1}{4}=\frac{3\sqrt{7}i}{4} x-\frac{1}{4}=-\frac{3\sqrt{7}i}{4}$$

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

$$x=\frac{1+3\sqrt{7}i}{4} x=\frac{-3\sqrt{7}i+1}{4}$$