1/4left(4^2+left(8-xright)^2+left(4+xright)^2right)=88 eching

$$\frac{ 1 }{ 4 } \left( { 4 }^{ 2 } + { \left(8-x \right) }^{ 2 } + { \left(4+x \right) }^{ 2 } \right) = 88$$

x uchun yechish

$$x=2\sqrt{33}+2\approx 13,489125293$$

$$x=2-2\sqrt{33}\approx -9,489125293$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikki tarafini 4 va teskari kasri \frac{1}{4} ga ko‘paytiring.

$$4^{2}+\left(8-x\right)^{2}+\left(4+x\right)^{2}=88\times 4$$

352 hosil qilish uchun 88 va 4 ni ko'paytirish.

$$4^{2}+\left(8-x\right)^{2}+\left(4+x\right)^{2}=352$$

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

$$16+\left(8-x\right)^{2}+\left(4+x\right)^{2}=352$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(8-x\right)^{2} kengaytirilishi uchun ishlating.

$$16+64-16x+x^{2}+\left(4+x\right)^{2}=352$$

80 olish uchun 16 va 64'ni qo'shing.

$$80-16x+x^{2}+\left(4+x\right)^{2}=352$$

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

$$80-16x+x^{2}+16+8x+x^{2}=352$$

96 olish uchun 80 va 16'ni qo'shing.

$$96-16x+x^{2}+8x+x^{2}=352$$

-8x ni olish uchun -16x va 8x ni birlashtirish.

$$96-8x+x^{2}+x^{2}=352$$

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

$$96-8x+2x^{2}=352$$

Ikkala tarafdan 352 ni ayirish.

$$96-8x+2x^{2}-352=0$$

-256 olish uchun 96 dan 352 ni ayirish.

$$-256-8x+2x^{2}=0$$

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.

$$2x^{2}-8x-256=0$$

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 -256 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\left(-256\right)}}{2\times 2}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 2\left(-256\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

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

-8 ni -256 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64+2048}}{2\times 2}$$

64 ni 2048 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{2112}}{2\times 2}$$

2112 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±8\sqrt{33}}{2\times 2}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±8\sqrt{33}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{8±8\sqrt{33}}{4}$$

x=\frac{8±8\sqrt{33}}{4} tenglamasini yeching, bunda ± musbat. 8 ni 8\sqrt{33} ga qo'shish.

$$x=\frac{8\sqrt{33}+8}{4}$$

8+8\sqrt{33} ni 4 ga bo'lish.

$$x=2\sqrt{33}+2$$

x=\frac{8±8\sqrt{33}}{4} tenglamasini yeching, bunda ± manfiy. 8 dan 8\sqrt{33} ni ayirish.

$$x=\frac{8-8\sqrt{33}}{4}$$

8-8\sqrt{33} ni 4 ga bo'lish.

$$x=2-2\sqrt{33}$$

Tenglama yechildi.

$$x=2\sqrt{33}+2 x=2-2\sqrt{33}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikki tarafini 4 va teskari kasri \frac{1}{4} ga ko‘paytiring.

$$4^{2}+\left(8-x\right)^{2}+\left(4+x\right)^{2}=88\times 4$$

352 hosil qilish uchun 88 va 4 ni ko'paytirish.

$$4^{2}+\left(8-x\right)^{2}+\left(4+x\right)^{2}=352$$

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

$$16+\left(8-x\right)^{2}+\left(4+x\right)^{2}=352$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(8-x\right)^{2} kengaytirilishi uchun ishlating.

$$16+64-16x+x^{2}+\left(4+x\right)^{2}=352$$

80 olish uchun 16 va 64'ni qo'shing.

$$80-16x+x^{2}+\left(4+x\right)^{2}=352$$

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

$$80-16x+x^{2}+16+8x+x^{2}=352$$

96 olish uchun 80 va 16'ni qo'shing.

$$96-16x+x^{2}+8x+x^{2}=352$$

-8x ni olish uchun -16x va 8x ni birlashtirish.

$$96-8x+x^{2}+x^{2}=352$$

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

$$96-8x+2x^{2}=352$$

Ikkala tarafdan 96 ni ayirish.

$$-8x+2x^{2}=352-96$$

256 olish uchun 352 dan 96 ni ayirish.

$$-8x+2x^{2}=256$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$2x^{2}-8x=256$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-8x}{2}=\frac{256}{2}$$

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

$$x^{2}+\left(-\frac{8}{2}\right)x=\frac{256}{2}$$

-8 ni 2 ga bo'lish.

$$x^{2}-4x=\frac{256}{2}$$

256 ni 2 ga bo'lish.

$$x^{2}-4x=128$$

-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.

$$x^{2}-4x+\left(-2\right)^{2}=128+\left(-2\right)^{2}$$

-2 kvadratini chiqarish.

$$x^{2}-4x+4=128+4$$

128 ni 4 ga qo'shish.

$$x^{2}-4x+4=132$$

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.

$$\left(x-2\right)^{2}=132$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{132}$$

Qisqartirish.

$$x-2=2\sqrt{33} x-2=-2\sqrt{33}$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=2\sqrt{33}+2 x=2-2\sqrt{33}$$