n(n+1)-2048=0 eching

$$n ( n + 1 ) - 2048 = 0$$

n uchun yechish

$$n = \frac{\sqrt{8193} - 1}{2} \approx 44,757596048$$

$$n=\frac{-\sqrt{8193}-1}{2}\approx -45,757596048$$

Kvadrat tenglamalaridan foydalanish qoidalari

n ga n+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$n^{2}+n-2048=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 1 ni b va -2048 ni c bilan almashtiring.

$$n=\frac{-1±\sqrt{1^{2}-4\left(-2048\right)}}{2}$$

1 kvadratini chiqarish.

$$n=\frac{-1±\sqrt{1-4\left(-2048\right)}}{2}$$

-4 ni -2048 marotabaga ko'paytirish.

$$n=\frac{-1±\sqrt{1+8192}}{2}$$

1 ni 8192 ga qo'shish.

$$n=\frac{-1±\sqrt{8193}}{2}$$

n=\frac{-1±\sqrt{8193}}{2} tenglamasini yeching, bunda ± musbat. -1 ni \sqrt{8193} ga qo'shish.

$$n=\frac{\sqrt{8193}-1}{2}$$

n=\frac{-1±\sqrt{8193}}{2} tenglamasini yeching, bunda ± manfiy. -1 dan \sqrt{8193} ni ayirish.

$$n=\frac{-\sqrt{8193}-1}{2}$$

Tenglama yechildi.

$$n=\frac{\sqrt{8193}-1}{2} n=\frac{-\sqrt{8193}-1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

n ga n+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$n^{2}+n-2048=0$$

2048 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$n^{2}+n=2048$$

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

$$n^{2}+n+\left(\frac{1}{2}\right)^{2}=2048+\left(\frac{1}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

$$n^{2}+n+\frac{1}{4}=2048+\frac{1}{4}$$

2048 ni \frac{1}{4} ga qo'shish.

$$n^{2}+n+\frac{1}{4}=\frac{8193}{4}$$

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

$$\left(n+\frac{1}{2}\right)^{2}=\frac{8193}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(n+\frac{1}{2}\right)^{2}}=\sqrt{\frac{8193}{4}}$$

Qisqartirish.

$$n+\frac{1}{2}=\frac{\sqrt{8193}}{2} n+\frac{1}{2}=-\frac{\sqrt{8193}}{2}$$

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$n=\frac{\sqrt{8193}-1}{2} n=\frac{-\sqrt{8193}-1}{2}$$