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

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

x uchun yechish (complex solution)

$$x=1+\sqrt{7}i\approx 1+2,645751311i$$

$$x=-\sqrt{7}i+1\approx 1-2,645751311i$$

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.

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

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

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

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

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

-2 ni 4 marotabaga ko'paytirish.

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

1 ni -8 ga qo'shish.

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

-7 ning kvadrat ildizini chiqarish.

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

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{7}i}{2\times \frac{1}{2}}$$

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

$$x=\frac{1±\sqrt{7}i}{1}$$

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

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

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

$$x=1+\sqrt{7}i$$

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

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

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

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

Tenglama yechildi.

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

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.

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

Tenglamaning ikkala tarafidan 4 ni ayirish.

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

O‘zidan 4 ayirilsa 0 qoladi.

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

Ikkala tarafini 2 ga ko‘paytiring.

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

\frac{1}{2} ga bo'lish \frac{1}{2} ga ko'paytirishni bekor qiladi.

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

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

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

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

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

-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.

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

-8 ni 1 ga qo'shish.

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

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.

$$\left(x-1\right)^{2}=-7$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-7}$$

Qisqartirish.

$$x-1=\sqrt{7}i x-1=-\sqrt{7}i$$

1 ni tenglamaning ikkala tarafiga qo'shish.

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