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2n^{2}-5n-4=6
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.
2n^{2}-5n-4-6=6-6
Tenglamaning ikkala tarafidan 6 ni ayirish.
2n^{2}-5n-4-6=0
O‘zidan 6 ayirilsa 0 qoladi.
2n^{2}-5n-10=0
-4 dan 6 ni ayirish.
n=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 2\left(-10\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, -5 ni b va -10 ni c bilan almashtiring.
n=\frac{-\left(-5\right)±\sqrt{25-4\times 2\left(-10\right)}}{2\times 2}
-5 kvadratini chiqarish.
n=\frac{-\left(-5\right)±\sqrt{25-8\left(-10\right)}}{2\times 2}
-4 ni 2 marotabaga ko'paytirish.
n=\frac{-\left(-5\right)±\sqrt{25+80}}{2\times 2}
-8 ni -10 marotabaga ko'paytirish.
n=\frac{-\left(-5\right)±\sqrt{105}}{2\times 2}
25 ni 80 ga qo'shish.
n=\frac{5±\sqrt{105}}{2\times 2}
-5 ning teskarisi 5 ga teng.
n=\frac{5±\sqrt{105}}{4}
2 ni 2 marotabaga ko'paytirish.
n=\frac{\sqrt{105}+5}{4}
n=\frac{5±\sqrt{105}}{4} tenglamasini yeching, bunda ± musbat. 5 ni \sqrt{105} ga qo'shish.
n=\frac{5-\sqrt{105}}{4}
n=\frac{5±\sqrt{105}}{4} tenglamasini yeching, bunda ± manfiy. 5 dan \sqrt{105} ni ayirish.
n=\frac{\sqrt{105}+5}{4} n=\frac{5-\sqrt{105}}{4}
Tenglama yechildi.
2n^{2}-5n-4=6
Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.
2n^{2}-5n-4-\left(-4\right)=6-\left(-4\right)
4 ni tenglamaning ikkala tarafiga qo'shish.
2n^{2}-5n=6-\left(-4\right)
O‘zidan -4 ayirilsa 0 qoladi.
2n^{2}-5n=10
6 dan -4 ni ayirish.
\frac{2n^{2}-5n}{2}=\frac{10}{2}
Ikki tarafini 2 ga bo‘ling.
n^{2}-\frac{5}{2}n=\frac{10}{2}
2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.
n^{2}-\frac{5}{2}n=5
10 ni 2 ga bo'lish.
n^{2}-\frac{5}{2}n+\left(-\frac{5}{4}\right)^{2}=5+\left(-\frac{5}{4}\right)^{2}
-\frac{5}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{4} olish uchun. Keyin, -\frac{5}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.
n^{2}-\frac{5}{2}n+\frac{25}{16}=5+\frac{25}{16}
Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{4} kvadratini chiqarish.
n^{2}-\frac{5}{2}n+\frac{25}{16}=\frac{105}{16}
5 ni \frac{25}{16} ga qo'shish.
\left(n-\frac{5}{4}\right)^{2}=\frac{105}{16}
n^{2}-\frac{5}{2}n+\frac{25}{16} 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(n-\frac{5}{4}\right)^{2}}=\sqrt{\frac{105}{16}}
Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.
n-\frac{5}{4}=\frac{\sqrt{105}}{4} n-\frac{5}{4}=-\frac{\sqrt{105}}{4}
Qisqartirish.
n=\frac{\sqrt{105}+5}{4} n=\frac{5-\sqrt{105}}{4}
\frac{5}{4} ni tenglamaning ikkala tarafiga qo'shish.