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x uchun yechish (complex solution)
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x^{2}-x+1=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.
x=\frac{-\left(-1\right)±\sqrt{1-4}}{2}
Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -1 ni b va 1 ni c bilan almashtiring.
x=\frac{-\left(-1\right)±\sqrt{-3}}{2}
1 ni -4 ga qo'shish.
x=\frac{-\left(-1\right)±\sqrt{3}i}{2}
-3 ning kvadrat ildizini chiqarish.
x=\frac{1±\sqrt{3}i}{2}
-1 ning teskarisi 1 ga teng.
x=\frac{1+\sqrt{3}i}{2}
x=\frac{1±\sqrt{3}i}{2} tenglamasini yeching, bunda ± musbat. 1 ni i\sqrt{3} ga qo'shish.
x=\frac{-\sqrt{3}i+1}{2}
x=\frac{1±\sqrt{3}i}{2} tenglamasini yeching, bunda ± manfiy. 1 dan i\sqrt{3} ni ayirish.
x=\frac{1+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i+1}{2}
Tenglama yechildi.
x^{2}-x+1=0
Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.
x^{2}-x+1-1=-1
Tenglamaning ikkala tarafidan 1 ni ayirish.
x^{2}-x=-1
O‘zidan 1 ayirilsa 0 qoladi.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-1+\left(-\frac{1}{2}\right)^{2}
-1 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{2} olish uchun. Keyin, -\frac{1}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.
x^{2}-x+\frac{1}{4}=-1+\frac{1}{4}
Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{2} kvadratini chiqarish.
x^{2}-x+\frac{1}{4}=-\frac{3}{4}
-1 ni \frac{1}{4} ga qo'shish.
\left(x-\frac{1}{2}\right)^{2}=-\frac{3}{4}
x^{2}-x+\frac{1}{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.
\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{3}{4}}
Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.
x-\frac{1}{2}=\frac{\sqrt{3}i}{2} x-\frac{1}{2}=-\frac{\sqrt{3}i}{2}
Qisqartirish.
x=\frac{1+\sqrt{3}i}{2} x=\frac{-\sqrt{3}i+1}{2}
\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.