2n^2-10n-5=-4n eching

$$2 n ^ { 2 } - 10 n - 5 = - 4 n$$

n uchun yechish

$$n = \frac{\sqrt{19} + 3}{2} \approx 3,679449472$$

$$n=\frac{3-\sqrt{19}}{2}\approx -0,679449472$$

Kvadrat tenglamalaridan foydalanish qoidalari

4n ni ikki tarafga qo’shing.

$$2n^{2}-10n-5+4n=0$$

-6n ni olish uchun -10n va 4n ni birlashtirish.

$$2n^{2}-6n-5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -6 ni b va -5 ni c bilan almashtiring.

$$n=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\left(-5\right)}}{2\times 2}$$

-6 kvadratini chiqarish.

$$n=\frac{-\left(-6\right)±\sqrt{36-4\times 2\left(-5\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$n=\frac{-\left(-6\right)±\sqrt{36-8\left(-5\right)}}{2\times 2}$$

-8 ni -5 marotabaga ko'paytirish.

$$n=\frac{-\left(-6\right)±\sqrt{36+40}}{2\times 2}$$

36 ni 40 ga qo'shish.

$$n=\frac{-\left(-6\right)±\sqrt{76}}{2\times 2}$$

76 ning kvadrat ildizini chiqarish.

$$n=\frac{-\left(-6\right)±2\sqrt{19}}{2\times 2}$$

-6 ning teskarisi 6 ga teng.

$$n=\frac{6±2\sqrt{19}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$n=\frac{6±2\sqrt{19}}{4}$$

n=\frac{6±2\sqrt{19}}{4} tenglamasini yeching, bunda ± musbat. 6 ni 2\sqrt{19} ga qo'shish.

$$n=\frac{2\sqrt{19}+6}{4}$$

6+2\sqrt{19} ni 4 ga bo'lish.

$$n=\frac{\sqrt{19}+3}{2}$$

n=\frac{6±2\sqrt{19}}{4} tenglamasini yeching, bunda ± manfiy. 6 dan 2\sqrt{19} ni ayirish.

$$n=\frac{6-2\sqrt{19}}{4}$$

6-2\sqrt{19} ni 4 ga bo'lish.

$$n=\frac{3-\sqrt{19}}{2}$$

Tenglama yechildi.

$$n=\frac{\sqrt{19}+3}{2} n=\frac{3-\sqrt{19}}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

4n ni ikki tarafga qo’shing.

$$2n^{2}-10n-5+4n=0$$

-6n ni olish uchun -10n va 4n ni birlashtirish.

$$2n^{2}-6n-5=0$$

5 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$2n^{2}-6n=5$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2n^{2}-6n}{2}=\frac{5}{2}$$

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

$$n^{2}+\left(-\frac{6}{2}\right)n=\frac{5}{2}$$

-6 ni 2 ga bo'lish.

$$n^{2}-3n=\frac{5}{2}$$

-3 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{2} olish uchun. Keyin, -\frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$n^{2}-3n+\left(-\frac{3}{2}\right)^{2}=\frac{5}{2}+\left(-\frac{3}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$n^{2}-3n+\frac{9}{4}=\frac{5}{2}+\frac{9}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{2} ni \frac{9}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$n^{2}-3n+\frac{9}{4}=\frac{19}{4}$$

n^{2}-3n+\frac{9}{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.

$$\left(n-\frac{3}{2}\right)^{2}=\frac{19}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(n-\frac{3}{2}\right)^{2}}=\sqrt{\frac{19}{4}}$$

Qisqartirish.

$$n-\frac{3}{2}=\frac{\sqrt{19}}{2} n-\frac{3}{2}=-\frac{\sqrt{19}}{2}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$n=\frac{\sqrt{19}+3}{2} n=\frac{3-\sqrt{19}}{2}$$