16+x^2+left(4-xright)^2+16={left(4sqrt{5}right)}^2 eching

$$16+ { x }^{ 2 } + { \left(4-x \right) }^{ 2 } +16= { \left(4 \sqrt{ 5 } \right) }^{ 2 }$$

x uchun yechish

$$x=2\sqrt{5}+2\approx 6,472135955$$

$$x=2-2\sqrt{5}\approx -2,472135955$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$16+x^{2}+16-8x+x^{2}+16=\left(4\sqrt{5}\right)^{2}$$

32 olish uchun 16 va 16'ni qo'shing.

$$32+x^{2}-8x+x^{2}+16=\left(4\sqrt{5}\right)^{2}$$

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

$$32+2x^{2}-8x+16=\left(4\sqrt{5}\right)^{2}$$

48 olish uchun 32 va 16'ni qo'shing.

$$48+2x^{2}-8x=\left(4\sqrt{5}\right)^{2}$$

\left(4\sqrt{5}\right)^{2} ni kengaytirish.

$$48+2x^{2}-8x=4^{2}\left(\sqrt{5}\right)^{2}$$

2 daraja ko‘rsatkichini 4 ga hisoblang va 16 ni qiymatni oling.

$$48+2x^{2}-8x=16\left(\sqrt{5}\right)^{2}$$

\sqrt{5} kvadrati – 5.

$$48+2x^{2}-8x=16\times 5$$

80 hosil qilish uchun 16 va 5 ni ko'paytirish.

$$48+2x^{2}-8x=80$$

Ikkala tarafdan 80 ni ayirish.

$$48+2x^{2}-8x-80=0$$

-32 olish uchun 48 dan 80 ni ayirish.

$$-32+2x^{2}-8x=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$2x^{2}-8x-32=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -8 ni b va -32 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\left(-32\right)}}{2\times 2}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 2\left(-32\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-8\left(-32\right)}}{2\times 2}$$

-8 ni -32 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64+256}}{2\times 2}$$

64 ni 256 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{320}}{2\times 2}$$

320 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±8\sqrt{5}}{2\times 2}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±8\sqrt{5}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{8±8\sqrt{5}}{4}$$

x=\frac{8±8\sqrt{5}}{4} tenglamasini yeching, bunda ± musbat. 8 ni 8\sqrt{5} ga qo'shish.

$$x=\frac{8\sqrt{5}+8}{4}$$

8+8\sqrt{5} ni 4 ga bo'lish.

$$x=2\sqrt{5}+2$$

x=\frac{8±8\sqrt{5}}{4} tenglamasini yeching, bunda ± manfiy. 8 dan 8\sqrt{5} ni ayirish.

$$x=\frac{8-8\sqrt{5}}{4}$$

8-8\sqrt{5} ni 4 ga bo'lish.

$$x=2-2\sqrt{5}$$

Tenglama yechildi.

$$x=2\sqrt{5}+2 x=2-2\sqrt{5}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$16+x^{2}+16-8x+x^{2}+16=\left(4\sqrt{5}\right)^{2}$$

32 olish uchun 16 va 16'ni qo'shing.

$$32+x^{2}-8x+x^{2}+16=\left(4\sqrt{5}\right)^{2}$$

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

$$32+2x^{2}-8x+16=\left(4\sqrt{5}\right)^{2}$$

48 olish uchun 32 va 16'ni qo'shing.

$$48+2x^{2}-8x=\left(4\sqrt{5}\right)^{2}$$

\left(4\sqrt{5}\right)^{2} ni kengaytirish.

$$48+2x^{2}-8x=4^{2}\left(\sqrt{5}\right)^{2}$$

2 daraja ko‘rsatkichini 4 ga hisoblang va 16 ni qiymatni oling.

$$48+2x^{2}-8x=16\left(\sqrt{5}\right)^{2}$$

\sqrt{5} kvadrati – 5.

$$48+2x^{2}-8x=16\times 5$$

80 hosil qilish uchun 16 va 5 ni ko'paytirish.

$$48+2x^{2}-8x=80$$

Ikkala tarafdan 48 ni ayirish.

$$2x^{2}-8x=80-48$$

32 olish uchun 80 dan 48 ni ayirish.

$$2x^{2}-8x=32$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-8x}{2}=\frac{32}{2}$$

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

$$x^{2}+\left(-\frac{8}{2}\right)x=\frac{32}{2}$$

-8 ni 2 ga bo'lish.

$$x^{2}-4x=\frac{32}{2}$$

32 ni 2 ga bo'lish.

$$x^{2}-4x=16$$

-4 ni bo‘lish, x shartining koeffitsienti, 2 ga -2 olish uchun. Keyin, -2 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-4x+\left(-2\right)^{2}=16+\left(-2\right)^{2}$$

-2 kvadratini chiqarish.

$$x^{2}-4x+4=16+4$$

16 ni 4 ga qo'shish.

$$x^{2}-4x+4=20$$

x^{2}-4x+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.

$$\left(x-2\right)^{2}=20$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{20}$$

Qisqartirish.

$$x-2=2\sqrt{5} x-2=-2\sqrt{5}$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=2\sqrt{5}+2 x=2-2\sqrt{5}$$