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4x^{2}-3x+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(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 4}}{2\times 4}
Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -3 ni b va 1 ni c bilan almashtiring.
x=\frac{-\left(-3\right)±\sqrt{9-4\times 4}}{2\times 4}
-3 kvadratini chiqarish.
x=\frac{-\left(-3\right)±\sqrt{9-16}}{2\times 4}
-4 ni 4 marotabaga ko'paytirish.
x=\frac{-\left(-3\right)±\sqrt{-7}}{2\times 4}
9 ni -16 ga qo'shish.
x=\frac{-\left(-3\right)±\sqrt{7}i}{2\times 4}
-7 ning kvadrat ildizini chiqarish.
x=\frac{3±\sqrt{7}i}{2\times 4}
-3 ning teskarisi 3 ga teng.
x=\frac{3±\sqrt{7}i}{8}
2 ni 4 marotabaga ko'paytirish.
x=\frac{3+\sqrt{7}i}{8}
x=\frac{3±\sqrt{7}i}{8} tenglamasini yeching, bunda ± musbat. 3 ni i\sqrt{7} ga qo'shish.
x=\frac{-\sqrt{7}i+3}{8}
x=\frac{3±\sqrt{7}i}{8} tenglamasini yeching, bunda ± manfiy. 3 dan i\sqrt{7} ni ayirish.
x=\frac{3+\sqrt{7}i}{8} x=\frac{-\sqrt{7}i+3}{8}
Tenglama yechildi.
4x^{2}-3x+1=0
Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.
4x^{2}-3x+1-1=-1
Tenglamaning ikkala tarafidan 1 ni ayirish.
4x^{2}-3x=-1
O‘zidan 1 ayirilsa 0 qoladi.
\frac{4x^{2}-3x}{4}=-\frac{1}{4}
Ikki tarafini 4 ga bo‘ling.
x^{2}-\frac{3}{4}x=-\frac{1}{4}
4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.
x^{2}-\frac{3}{4}x+\left(-\frac{3}{8}\right)^{2}=-\frac{1}{4}+\left(-\frac{3}{8}\right)^{2}
-\frac{3}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{8} olish uchun. Keyin, -\frac{3}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.
x^{2}-\frac{3}{4}x+\frac{9}{64}=-\frac{1}{4}+\frac{9}{64}
Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{8} kvadratini chiqarish.
x^{2}-\frac{3}{4}x+\frac{9}{64}=-\frac{7}{64}
Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{4} ni \frac{9}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.
\left(x-\frac{3}{8}\right)^{2}=-\frac{7}{64}
x^{2}-\frac{3}{4}x+\frac{9}{64} 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{3}{8}\right)^{2}}=\sqrt{-\frac{7}{64}}
Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.
x-\frac{3}{8}=\frac{\sqrt{7}i}{8} x-\frac{3}{8}=-\frac{\sqrt{7}i}{8}
Qisqartirish.
x=\frac{3+\sqrt{7}i}{8} x=\frac{-\sqrt{7}i+3}{8}
\frac{3}{8} ni tenglamaning ikkala tarafiga qo'shish.