(x+4)^2+(x-5)^2=2^2 eching

$$( x + 4 ) ^ { 2 } + ( x - 5 ) ^ { 2 } = 2 ^ { 2 }$$

x uchun yechish (complex solution)

$$x=\frac{1+\sqrt{73}i}{2}\approx 0,5+4,272001873i$$

$$x=\frac{-\sqrt{73}i+1}{2}\approx 0,5-4,272001873i$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+4\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+8x+16+\left(x-5\right)^{2}=2^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-5\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+8x+16+x^{2}-10x+25=2^{2}$$

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

$$2x^{2}+8x+16-10x+25=2^{2}$$

-2x ni olish uchun 8x va -10x ni birlashtirish.

$$2x^{2}-2x+16+25=2^{2}$$

41 olish uchun 16 va 25'ni qo'shing.

$$2x^{2}-2x+41=2^{2}$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$2x^{2}-2x+41=4$$

Ikkala tarafdan 4 ni ayirish.

$$2x^{2}-2x+41-4=0$$

37 olish uchun 41 dan 4 ni ayirish.

$$2x^{2}-2x+37=0$$

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 37 ni c bilan almashtiring.

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

-2 kvadratini chiqarish.

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

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-8\times 37}}{2\times 2}$$

-8 ni 37 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-296}}{2\times 2}$$

4 ni -296 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{-292}}{2\times 2}$$

-292 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±2\sqrt{73}i}{2\times 2}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±2\sqrt{73}i}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{2±2\sqrt{73}i}{4}$$

x=\frac{2±2\sqrt{73}i}{4} tenglamasini yeching, bunda ± musbat. 2 ni 2i\sqrt{73} ga qo'shish.

$$x=\frac{2+2\sqrt{73}i}{4}$$

2+2i\sqrt{73} ni 4 ga bo'lish.

$$x=\frac{1+\sqrt{73}i}{2}$$

x=\frac{2±2\sqrt{73}i}{4} tenglamasini yeching, bunda ± manfiy. 2 dan 2i\sqrt{73} ni ayirish.

$$x=\frac{-2\sqrt{73}i+2}{4}$$

2-2i\sqrt{73} ni 4 ga bo'lish.

$$x=\frac{-\sqrt{73}i+1}{2}$$

Tenglama yechildi.

$$x=\frac{1+\sqrt{73}i}{2} x=\frac{-\sqrt{73}i+1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+4\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+8x+16+\left(x-5\right)^{2}=2^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-5\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+8x+16+x^{2}-10x+25=2^{2}$$

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

$$2x^{2}+8x+16-10x+25=2^{2}$$

-2x ni olish uchun 8x va -10x ni birlashtirish.

$$2x^{2}-2x+16+25=2^{2}$$

41 olish uchun 16 va 25'ni qo'shing.

$$2x^{2}-2x+41=2^{2}$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$2x^{2}-2x+41=4$$

Ikkala tarafdan 41 ni ayirish.

$$2x^{2}-2x=4-41$$

-37 olish uchun 4 dan 41 ni ayirish.

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

Ikki tarafini 2 ga bo‘ling.

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

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

$$x^{2}+\left(-\frac{2}{2}\right)x=-\frac{37}{2}$$

-2 ni 2 ga bo'lish.

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

-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.

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

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

$$x^{2}-x+\frac{1}{4}=-\frac{37}{2}+\frac{1}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{37}{2} ni \frac{1}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-x+\frac{1}{4}=-\frac{73}{4}$$

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.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{73}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{73}{4}}$$

Qisqartirish.

$$x-\frac{1}{2}=\frac{\sqrt{73}i}{2} x-\frac{1}{2}=-\frac{\sqrt{73}i}{2}$$

\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{73}i}{2} x=\frac{-\sqrt{73}i+1}{2}$$