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Baham ko'rish

3x^{2}-8x-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(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 3\left(-1\right)}}{2\times 3}
Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -8 ni b va -1 ni c bilan almashtiring.
x=\frac{-\left(-8\right)±\sqrt{64-4\times 3\left(-1\right)}}{2\times 3}
-8 kvadratini chiqarish.
x=\frac{-\left(-8\right)±\sqrt{64-12\left(-1\right)}}{2\times 3}
-4 ni 3 marotabaga ko'paytirish.
x=\frac{-\left(-8\right)±\sqrt{64+12}}{2\times 3}
-12 ni -1 marotabaga ko'paytirish.
x=\frac{-\left(-8\right)±\sqrt{76}}{2\times 3}
64 ni 12 ga qo'shish.
x=\frac{-\left(-8\right)±2\sqrt{19}}{2\times 3}
76 ning kvadrat ildizini chiqarish.
x=\frac{8±2\sqrt{19}}{2\times 3}
-8 ning teskarisi 8 ga teng.
x=\frac{8±2\sqrt{19}}{6}
2 ni 3 marotabaga ko'paytirish.
x=\frac{2\sqrt{19}+8}{6}
x=\frac{8±2\sqrt{19}}{6} tenglamasini yeching, bunda ± musbat. 8 ni 2\sqrt{19} ga qo'shish.
x=\frac{\sqrt{19}+4}{3}
8+2\sqrt{19} ni 6 ga bo'lish.
x=\frac{8-2\sqrt{19}}{6}
x=\frac{8±2\sqrt{19}}{6} tenglamasini yeching, bunda ± manfiy. 8 dan 2\sqrt{19} ni ayirish.
x=\frac{4-\sqrt{19}}{3}
8-2\sqrt{19} ni 6 ga bo'lish.
x=\frac{\sqrt{19}+4}{3} x=\frac{4-\sqrt{19}}{3}
Tenglama yechildi.
3x^{2}-8x-1=0
Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.
3x^{2}-8x-1-\left(-1\right)=-\left(-1\right)
1 ni tenglamaning ikkala tarafiga qo'shish.
3x^{2}-8x=-\left(-1\right)
O‘zidan -1 ayirilsa 0 qoladi.
3x^{2}-8x=1
0 dan -1 ni ayirish.
\frac{3x^{2}-8x}{3}=\frac{1}{3}
Ikki tarafini 3 ga bo‘ling.
x^{2}-\frac{8}{3}x=\frac{1}{3}
3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.
x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=\frac{1}{3}+\left(-\frac{4}{3}\right)^{2}
-\frac{8}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{4}{3} olish uchun. Keyin, -\frac{4}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.
x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{1}{3}+\frac{16}{9}
Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{4}{3} kvadratini chiqarish.
x^{2}-\frac{8}{3}x+\frac{16}{9}=\frac{19}{9}
Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{3} ni \frac{16}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.
\left(x-\frac{4}{3}\right)^{2}=\frac{19}{9}
x^{2}-\frac{8}{3}x+\frac{16}{9} 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{4}{3}\right)^{2}}=\sqrt{\frac{19}{9}}
Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.
x-\frac{4}{3}=\frac{\sqrt{19}}{3} x-\frac{4}{3}=-\frac{\sqrt{19}}{3}
Qisqartirish.
x=\frac{\sqrt{19}+4}{3} x=\frac{4-\sqrt{19}}{3}
\frac{4}{3} ni tenglamaning ikkala tarafiga qo'shish.