(x+43)^2+(2x+34-8)^2=0 eching

$$( x + 43 ) ^ { 2 } + ( 2 x + 34 - 8 ) ^ { 2 } = 0$$

x uchun yechish (complex solution)

$$x=-19+12i$$

$$x=-19-12i$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+43\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+86x+1849+\left(2x+34-8\right)^{2}=0$$

26 olish uchun 34 dan 8 ni ayirish.

$$x^{2}+86x+1849+\left(2x+26\right)^{2}=0$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+26\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+86x+1849+4x^{2}+104x+676=0$$

5x^{2} ni olish uchun x^{2} va 4x^{2} ni birlashtirish.

$$5x^{2}+86x+1849+104x+676=0$$

190x ni olish uchun 86x va 104x ni birlashtirish.

$$5x^{2}+190x+1849+676=0$$

2525 olish uchun 1849 va 676'ni qo'shing.

$$5x^{2}+190x+2525=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, 190 ni b va 2525 ni c bilan almashtiring.

$$x=\frac{-190±\sqrt{190^{2}-4\times 5\times 2525}}{2\times 5}$$

190 kvadratini chiqarish.

$$x=\frac{-190±\sqrt{36100-4\times 5\times 2525}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-190±\sqrt{36100-20\times 2525}}{2\times 5}$$

-20 ni 2525 marotabaga ko'paytirish.

$$x=\frac{-190±\sqrt{36100-50500}}{2\times 5}$$

36100 ni -50500 ga qo'shish.

$$x=\frac{-190±\sqrt{-14400}}{2\times 5}$$

-14400 ning kvadrat ildizini chiqarish.

$$x=\frac{-190±120i}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{-190±120i}{10}$$

x=\frac{-190±120i}{10} tenglamasini yeching, bunda ± musbat. -190 ni 120i ga qo'shish.

$$x=\frac{-190+120i}{10}$$

-190+120i ni 10 ga bo'lish.

$$x=-19+12i$$

x=\frac{-190±120i}{10} tenglamasini yeching, bunda ± manfiy. -190 dan 120i ni ayirish.

$$x=\frac{-190-120i}{10}$$

-190-120i ni 10 ga bo'lish.

$$x=-19-12i$$

Tenglama yechildi.

$$x=-19+12i x=-19-12i$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+43\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+86x+1849+\left(2x+34-8\right)^{2}=0$$

26 olish uchun 34 dan 8 ni ayirish.

$$x^{2}+86x+1849+\left(2x+26\right)^{2}=0$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+26\right)^{2} kengaytirilishi uchun ishlating.

$$x^{2}+86x+1849+4x^{2}+104x+676=0$$

5x^{2} ni olish uchun x^{2} va 4x^{2} ni birlashtirish.

$$5x^{2}+86x+1849+104x+676=0$$

190x ni olish uchun 86x va 104x ni birlashtirish.

$$5x^{2}+190x+1849+676=0$$

2525 olish uchun 1849 va 676'ni qo'shing.

$$5x^{2}+190x+2525=0$$

Ikkala tarafdan 2525 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$5x^{2}+190x=-2525$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}+190x}{5}=-\frac{2525}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{190}{5}x=-\frac{2525}{5}$$

190 ni 5 ga bo'lish.

$$x^{2}+38x=-\frac{2525}{5}$$

-2525 ni 5 ga bo'lish.

$$x^{2}+38x=-505$$

38 ni bo‘lish, x shartining koeffitsienti, 2 ga 19 olish uchun. Keyin, 19 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+38x+19^{2}=-505+19^{2}$$

19 kvadratini chiqarish.

$$x^{2}+38x+361=-505+361$$

-505 ni 361 ga qo'shish.

$$x^{2}+38x+361=-144$$

x^{2}+38x+361 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+19\right)^{2}=-144$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+19\right)^{2}}=\sqrt{-144}$$

Qisqartirish.

$$x+19=12i x+19=-12i$$

Tenglamaning ikkala tarafidan 19 ni ayirish.

$$x=-19+12i x=-19-12i$$