2n^2-n-37=0 eching

$$2 n ^ { 2 } - n - 37 = 0$$

n uchun yechish

$$n = \frac{3 \sqrt{33} + 1}{4} \approx 4,558421985$$

$$n=\frac{1-3\sqrt{33}}{4}\approx -4,058421985$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$2n^{2}-n-37=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -1 ni b va -37 ni c bilan almashtiring.

$$n=\frac{-\left(-1\right)±\sqrt{1-4\times 2\left(-37\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$n=\frac{-\left(-1\right)±\sqrt{1-8\left(-37\right)}}{2\times 2}$$

-8 ni -37 marotabaga ko'paytirish.

$$n=\frac{-\left(-1\right)±\sqrt{1+296}}{2\times 2}$$

1 ni 296 ga qo'shish.

$$n=\frac{-\left(-1\right)±\sqrt{297}}{2\times 2}$$

297 ning kvadrat ildizini chiqarish.

$$n=\frac{-\left(-1\right)±3\sqrt{33}}{2\times 2}$$

-1 ning teskarisi 1 ga teng.

$$n=\frac{1±3\sqrt{33}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$n=\frac{1±3\sqrt{33}}{4}$$

n=\frac{1±3\sqrt{33}}{4} tenglamasini yeching, bunda ± musbat. 1 ni 3\sqrt{33} ga qo'shish.

$$n=\frac{3\sqrt{33}+1}{4}$$

n=\frac{1±3\sqrt{33}}{4} tenglamasini yeching, bunda ± manfiy. 1 dan 3\sqrt{33} ni ayirish.

$$n=\frac{1-3\sqrt{33}}{4}$$

Tenglama yechildi.

$$n=\frac{3\sqrt{33}+1}{4} n=\frac{1-3\sqrt{33}}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$2n^{2}-n-37=0$$

37 ni tenglamaning ikkala tarafiga qo'shish.

$$2n^{2}-n-37-\left(-37\right)=-\left(-37\right)$$

O‘zidan -37 ayirilsa 0 qoladi.

$$2n^{2}-n=-\left(-37\right)$$

0 dan -37 ni ayirish.

$$2n^{2}-n=37$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2n^{2}-n}{2}=\frac{37}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$n^{2}-\frac{1}{2}n=\frac{37}{2}$$

-\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{4} olish uchun. Keyin, -\frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$n^{2}-\frac{1}{2}n+\left(-\frac{1}{4}\right)^{2}=\frac{37}{2}+\left(-\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{4} kvadratini chiqarish.

$$n^{2}-\frac{1}{2}n+\frac{1}{16}=\frac{37}{2}+\frac{1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{37}{2} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$n^{2}-\frac{1}{2}n+\frac{1}{16}=\frac{297}{16}$$

n^{2}-\frac{1}{2}n+\frac{1}{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.

$$\left(n-\frac{1}{4}\right)^{2}=\frac{297}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(n-\frac{1}{4}\right)^{2}}=\sqrt{\frac{297}{16}}$$

Qisqartirish.

$$n-\frac{1}{4}=\frac{3\sqrt{33}}{4} n-\frac{1}{4}=-\frac{3\sqrt{33}}{4}$$

\frac{1}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$n=\frac{3\sqrt{33}+1}{4} n=\frac{1-3\sqrt{33}}{4}$$