2^152m^2-2m-1=0 eching

$$2 ^ { 1 } 52 m ^ { 2 } - 2 m - 1 = 0$$

m uchun yechish

$$m=\frac{\sqrt{105}+1}{104}\approx 0,108143757$$

$$m=\frac{1-\sqrt{105}}{104}\approx -0,088912988$$

Kvadrat tenglamalaridan foydalanish qoidalari

1 daraja ko‘rsatkichini 2 ga hisoblang va 2 ni qiymatni oling.

$$2\times 52m^{2}-2m-1=0$$

104 hosil qilish uchun 2 va 52 ni ko'paytirish.

$$104m^{2}-2m-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 104 ni a, -2 ni b va -1 ni c bilan almashtiring.

$$m=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 104\left(-1\right)}}{2\times 104}$$

-2 kvadratini chiqarish.

$$m=\frac{-\left(-2\right)±\sqrt{4-4\times 104\left(-1\right)}}{2\times 104}$$

-4 ni 104 marotabaga ko'paytirish.

$$m=\frac{-\left(-2\right)±\sqrt{4-416\left(-1\right)}}{2\times 104}$$

-416 ni -1 marotabaga ko'paytirish.

$$m=\frac{-\left(-2\right)±\sqrt{4+416}}{2\times 104}$$

4 ni 416 ga qo'shish.

$$m=\frac{-\left(-2\right)±\sqrt{420}}{2\times 104}$$

420 ning kvadrat ildizini chiqarish.

$$m=\frac{-\left(-2\right)±2\sqrt{105}}{2\times 104}$$

-2 ning teskarisi 2 ga teng.

$$m=\frac{2±2\sqrt{105}}{2\times 104}$$

2 ni 104 marotabaga ko'paytirish.

$$m=\frac{2±2\sqrt{105}}{208}$$

m=\frac{2±2\sqrt{105}}{208} tenglamasini yeching, bunda ± musbat. 2 ni 2\sqrt{105} ga qo'shish.

$$m=\frac{2\sqrt{105}+2}{208}$$

2+2\sqrt{105} ni 208 ga bo'lish.

$$m=\frac{\sqrt{105}+1}{104}$$

m=\frac{2±2\sqrt{105}}{208} tenglamasini yeching, bunda ± manfiy. 2 dan 2\sqrt{105} ni ayirish.

$$m=\frac{2-2\sqrt{105}}{208}$$

2-2\sqrt{105} ni 208 ga bo'lish.

$$m=\frac{1-\sqrt{105}}{104}$$

Tenglama yechildi.

$$m=\frac{\sqrt{105}+1}{104} m=\frac{1-\sqrt{105}}{104}$$

Kvadratni yakunlash bo'yicha qadamlar

1 daraja ko‘rsatkichini 2 ga hisoblang va 2 ni qiymatni oling.

$$2\times 52m^{2}-2m-1=0$$

104 hosil qilish uchun 2 va 52 ni ko'paytirish.

$$104m^{2}-2m-1=0$$

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

$$104m^{2}-2m=1$$

Ikki tarafini 104 ga bo‘ling.

$$\frac{104m^{2}-2m}{104}=\frac{1}{104}$$

104 ga bo'lish 104 ga ko'paytirishni bekor qiladi.

$$m^{2}+\left(-\frac{2}{104}\right)m=\frac{1}{104}$$

\frac{-2}{104} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$m^{2}-\frac{1}{52}m=\frac{1}{104}$$

-\frac{1}{52} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{104} olish uchun. Keyin, -\frac{1}{104} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$m^{2}-\frac{1}{52}m+\left(-\frac{1}{104}\right)^{2}=\frac{1}{104}+\left(-\frac{1}{104}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{104} kvadratini chiqarish.

$$m^{2}-\frac{1}{52}m+\frac{1}{10816}=\frac{1}{104}+\frac{1}{10816}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{104} ni \frac{1}{10816} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$m^{2}-\frac{1}{52}m+\frac{1}{10816}=\frac{105}{10816}$$

m^{2}-\frac{1}{52}m+\frac{1}{10816} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(m-\frac{1}{104}\right)^{2}=\frac{105}{10816}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(m-\frac{1}{104}\right)^{2}}=\sqrt{\frac{105}{10816}}$$

Qisqartirish.

$$m-\frac{1}{104}=\frac{\sqrt{105}}{104} m-\frac{1}{104}=-\frac{\sqrt{105}}{104}$$

\frac{1}{104} ni tenglamaning ikkala tarafiga qo'shish.

$$m=\frac{\sqrt{105}+1}{104} m=\frac{1-\sqrt{105}}{104}$$