x^2+(x-2)^2=100 eching

$$x ^ { 2 } + ( x - 2 ) ^ { 2 } = 100$$

x uchun yechish

$$x=-6$$

$$x=8$$

Guruhlash orqali faktorlash qadamlari

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

$$x^{2}+x^{2}-4x+4=100$$

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

$$2x^{2}-4x+4=100$$

Ikkala tarafdan 100 ni ayirish.

$$2x^{2}-4x+4-100=0$$

-96 olish uchun 4 dan 100 ni ayirish.

$$2x^{2}-4x-96=0$$

Ikki tarafini 2 ga bo‘ling.

$$x^{2}-2x-48=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx-48 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-2 ab=1\left(-48\right)=-48$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -48-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Har bir juftlik yigʻindisini hisoblang.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

Yechim – -2 yigʻindisini beruvchi juftlik.

$$a=-8 b=6$$

x^{2}-2x-48 ni \left(x^{2}-8x\right)+\left(6x-48\right) sifatida qaytadan yozish.

$$\left(x^{2}-8x\right)+\left(6x-48\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 6 ni faktordan chiqaring.

$$x\left(x-8\right)+6\left(x-8\right)$$

Distributiv funktsiyasidan foydalangan holda x-8 umumiy terminini chiqaring.

$$\left(x-8\right)\left(x+6\right)$$

Tenglamani yechish uchun x-8=0 va x+6=0 ni yeching.

$$x=8 x=-6$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$x^{2}+x^{2}-4x+4=100$$

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

$$2x^{2}-4x+4=100$$

Ikkala tarafdan 100 ni ayirish.

$$2x^{2}-4x+4-100=0$$

-96 olish uchun 4 dan 100 ni ayirish.

$$2x^{2}-4x-96=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -4 ni b va -96 ni c bilan almashtiring.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-96\right)}}{2\times 2}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-96\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-96\right)}}{2\times 2}$$

-8 ni -96 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16+768}}{2\times 2}$$

16 ni 768 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{784}}{2\times 2}$$

784 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±28}{2\times 2}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±28}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{4±28}{4}$$

x=\frac{4±28}{4} tenglamasini yeching, bunda ± musbat. 4 ni 28 ga qo'shish.

$$x=\frac{32}{4}$$

32 ni 4 ga bo'lish.

$$x=8$$

x=\frac{4±28}{4} tenglamasini yeching, bunda ± manfiy. 4 dan 28 ni ayirish.

$$x=-\frac{24}{4}$$

-24 ni 4 ga bo'lish.

$$x=-6$$

Tenglama yechildi.

$$x=8 x=-6$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$x^{2}+x^{2}-4x+4=100$$

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

$$2x^{2}-4x+4=100$$

Ikkala tarafdan 4 ni ayirish.

$$2x^{2}-4x=100-4$$

96 olish uchun 100 dan 4 ni ayirish.

$$2x^{2}-4x=96$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-4x}{2}=\frac{96}{2}$$

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

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{96}{2}$$

-4 ni 2 ga bo'lish.

$$x^{2}-2x=\frac{96}{2}$$

96 ni 2 ga bo'lish.

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

-2 ni bo‘lish, x shartining koeffitsienti, 2 ga -1 olish uchun. Keyin, -1 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-2x+1=48+1$$

48 ni 1 ga qo'shish.

$$x^{2}-2x+1=49$$

x^{2}-2x+1 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-1\right)^{2}=49$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{49}$$

Qisqartirish.

$$x-1=7 x-1=-7$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$x=8 x=-6$$