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2x^{2}-2x=1
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}-2x-1=1-1
Tenglamaning ikkala tarafidan 1 ni ayirish.
2x^{2}-2x-1=0
O‘zidan 1 ayirilsa 0 qoladi.
x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-1\right)}}{2\times 2}
Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -2 ni b va -1 ni c bilan almashtiring.
x=\frac{-\left(-2\right)±\sqrt{4-4\times 2\left(-1\right)}}{2\times 2}
-2 kvadratini chiqarish.
x=\frac{-\left(-2\right)±\sqrt{4-8\left(-1\right)}}{2\times 2}
-4 ni 2 marotabaga ko'paytirish.
x=\frac{-\left(-2\right)±\sqrt{4+8}}{2\times 2}
-8 ni -1 marotabaga ko'paytirish.
x=\frac{-\left(-2\right)±\sqrt{12}}{2\times 2}
4 ni 8 ga qo'shish.
x=\frac{-\left(-2\right)±2\sqrt{3}}{2\times 2}
12 ning kvadrat ildizini chiqarish.
x=\frac{2±2\sqrt{3}}{2\times 2}
-2 ning teskarisi 2 ga teng.
x=\frac{2±2\sqrt{3}}{4}
2 ni 2 marotabaga ko'paytirish.
x=\frac{2\sqrt{3}+2}{4}
x=\frac{2±2\sqrt{3}}{4} tenglamasini yeching, bunda ± musbat. 2 ni 2\sqrt{3} ga qo'shish.
x=\frac{\sqrt{3}+1}{2}
2+2\sqrt{3} ni 4 ga bo'lish.
x=\frac{2-2\sqrt{3}}{4}
x=\frac{2±2\sqrt{3}}{4} tenglamasini yeching, bunda ± manfiy. 2 dan 2\sqrt{3} ni ayirish.
x=\frac{1-\sqrt{3}}{2}
2-2\sqrt{3} ni 4 ga bo'lish.
x=\frac{\sqrt{3}+1}{2} x=\frac{1-\sqrt{3}}{2}
Tenglama yechildi.
2x^{2}-2x=1
Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.
\frac{2x^{2}-2x}{2}=\frac{1}{2}
Ikki tarafini 2 ga bo‘ling.
x^{2}+\left(-\frac{2}{2}\right)x=\frac{1}{2}
2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.
x^{2}-x=\frac{1}{2}
-2 ni 2 ga bo'lish.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{1}{2}+\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}=\frac{1}{2}+\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}
Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{1}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.
\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}}{2} x-\frac{1}{2}=-\frac{\sqrt{3}}{2}
Qisqartirish.
x=\frac{\sqrt{3}+1}{2} x=\frac{1-\sqrt{3}}{2}
\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.