x^2+(x+1)^2=4191 eching

$$x ^ { 2 } + ( x + 1 ) ^ { 2 } = 4191$$

x uchun yechish

$$x = \frac{17 \sqrt{29} - 1}{2} \approx 45,273900861$$

$$x=\frac{-17\sqrt{29}-1}{2}\approx -46,273900861$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+x^{2}+2x+1=4191$$

2x^{2} ni olish uchun x^{2} va x^{2} ni birlashtirish.

$$2x^{2}+2x+1=4191$$

Ikkala tarafdan 4191 ni ayirish.

$$2x^{2}+2x+1-4191=0$$

-4190 olish uchun 1 dan 4191 ni ayirish.

$$2x^{2}+2x-4190=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 2 ni b va -4190 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\times 2\left(-4190\right)}}{2\times 2}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\times 2\left(-4190\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-8\left(-4190\right)}}{2\times 2}$$

-8 ni -4190 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+33520}}{2\times 2}$$

4 ni 33520 ga qo'shish.

$$x=\frac{-2±\sqrt{33524}}{2\times 2}$$

33524 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±34\sqrt{29}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-2±34\sqrt{29}}{4}$$

x=\frac{-2±34\sqrt{29}}{4} tenglamasini yeching, bunda ± musbat. -2 ni 34\sqrt{29} ga qo'shish.

$$x=\frac{34\sqrt{29}-2}{4}$$

-2+34\sqrt{29} ni 4 ga bo'lish.

$$x=\frac{17\sqrt{29}-1}{2}$$

x=\frac{-2±34\sqrt{29}}{4} tenglamasini yeching, bunda ± manfiy. -2 dan 34\sqrt{29} ni ayirish.

$$x=\frac{-34\sqrt{29}-2}{4}$$

-2-34\sqrt{29} ni 4 ga bo'lish.

$$x=\frac{-17\sqrt{29}-1}{2}$$

Tenglama yechildi.

$$x=\frac{17\sqrt{29}-1}{2} x=\frac{-17\sqrt{29}-1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+x^{2}+2x+1=4191$$

2x^{2} ni olish uchun x^{2} va x^{2} ni birlashtirish.

$$2x^{2}+2x+1=4191$$

Ikkala tarafdan 1 ni ayirish.

$$2x^{2}+2x=4191-1$$

4190 olish uchun 4191 dan 1 ni ayirish.

$$2x^{2}+2x=4190$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+2x}{2}=\frac{4190}{2}$$

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

$$x^{2}+\frac{2}{2}x=\frac{4190}{2}$$

2 ni 2 ga bo'lish.

$$x^{2}+x=\frac{4190}{2}$$

4190 ni 2 ga bo'lish.

$$x^{2}+x=2095$$

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.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=2095+\left(\frac{1}{2}\right)^{2}$$

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

$$x^{2}+x+\frac{1}{4}=2095+\frac{1}{4}$$

2095 ni \frac{1}{4} ga qo'shish.

$$x^{2}+x+\frac{1}{4}=\frac{8381}{4}$$

x^{2}+x+\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.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{8381}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{8381}{4}}$$

Qisqartirish.

$$x+\frac{1}{2}=\frac{17\sqrt{29}}{2} x+\frac{1}{2}=-\frac{17\sqrt{29}}{2}$$

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

$$x=\frac{17\sqrt{29}-1}{2} x=\frac{-17\sqrt{29}-1}{2}$$