21left(x-2right)^2-(x-2)=2 eching

$$21 { \left(x-2 \right) }^{ 2 } -(x-2)=2$$

x uchun yechish

$$x = \frac{12}{7} = 1\frac{5}{7} \approx 1,714285714$$

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2,333333333$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$21\left(x^{2}-4x+4\right)-\left(x-2\right)=2$$

21 ga x^{2}-4x+4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$21x^{2}-84x+84-\left(x-2\right)=2$$

x-2 teskarisini topish uchun har birining teskarisini toping.

$$21x^{2}-84x+84-x+2=2$$

-85x ni olish uchun -84x va -x ni birlashtirish.

$$21x^{2}-85x+84+2=2$$

86 olish uchun 84 va 2'ni qo'shing.

$$21x^{2}-85x+86=2$$

Ikkala tarafdan 2 ni ayirish.

$$21x^{2}-85x+86-2=0$$

84 olish uchun 86 dan 2 ni ayirish.

$$21x^{2}-85x+84=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 21 ni a, -85 ni b va 84 ni c bilan almashtiring.

$$x=\frac{-\left(-85\right)±\sqrt{\left(-85\right)^{2}-4\times 21\times 84}}{2\times 21}$$

-85 kvadratini chiqarish.

$$x=\frac{-\left(-85\right)±\sqrt{7225-4\times 21\times 84}}{2\times 21}$$

-4 ni 21 marotabaga ko'paytirish.

$$x=\frac{-\left(-85\right)±\sqrt{7225-84\times 84}}{2\times 21}$$

-84 ni 84 marotabaga ko'paytirish.

$$x=\frac{-\left(-85\right)±\sqrt{7225-7056}}{2\times 21}$$

7225 ni -7056 ga qo'shish.

$$x=\frac{-\left(-85\right)±\sqrt{169}}{2\times 21}$$

169 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-85\right)±13}{2\times 21}$$

-85 ning teskarisi 85 ga teng.

$$x=\frac{85±13}{2\times 21}$$

2 ni 21 marotabaga ko'paytirish.

$$x=\frac{85±13}{42}$$

x=\frac{85±13}{42} tenglamasini yeching, bunda ± musbat. 85 ni 13 ga qo'shish.

$$x=\frac{98}{42}$$

\frac{98}{42} ulushini 14 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{7}{3}$$

x=\frac{85±13}{42} tenglamasini yeching, bunda ± manfiy. 85 dan 13 ni ayirish.

$$x=\frac{72}{42}$$

\frac{72}{42} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{12}{7}$$

Tenglama yechildi.

$$x=\frac{7}{3} x=\frac{12}{7}$$

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.

$$21\left(x^{2}-4x+4\right)-\left(x-2\right)=2$$

21 ga x^{2}-4x+4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$21x^{2}-84x+84-\left(x-2\right)=2$$

x-2 teskarisini topish uchun har birining teskarisini toping.

$$21x^{2}-84x+84-x+2=2$$

-85x ni olish uchun -84x va -x ni birlashtirish.

$$21x^{2}-85x+84+2=2$$

86 olish uchun 84 va 2'ni qo'shing.

$$21x^{2}-85x+86=2$$

Ikkala tarafdan 86 ni ayirish.

$$21x^{2}-85x=2-86$$

-84 olish uchun 2 dan 86 ni ayirish.

$$21x^{2}-85x=-84$$

Ikki tarafini 21 ga bo‘ling.

$$\frac{21x^{2}-85x}{21}=-\frac{84}{21}$$

21 ga bo'lish 21 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{85}{21}x=-\frac{84}{21}$$

-84 ni 21 ga bo'lish.

$$x^{2}-\frac{85}{21}x=-4$$

-\frac{85}{21} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{85}{42} olish uchun. Keyin, -\frac{85}{42} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{85}{21}x+\left(-\frac{85}{42}\right)^{2}=-4+\left(-\frac{85}{42}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{85}{42} kvadratini chiqarish.

$$x^{2}-\frac{85}{21}x+\frac{7225}{1764}=-4+\frac{7225}{1764}$$

-4 ni \frac{7225}{1764} ga qo'shish.

$$x^{2}-\frac{85}{21}x+\frac{7225}{1764}=\frac{169}{1764}$$

x^{2}-\frac{85}{21}x+\frac{7225}{1764} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{85}{42}\right)^{2}=\frac{169}{1764}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{85}{42}\right)^{2}}=\sqrt{\frac{169}{1764}}$$

Qisqartirish.

$$x-\frac{85}{42}=\frac{13}{42} x-\frac{85}{42}=-\frac{13}{42}$$

\frac{85}{42} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{7}{3} x=\frac{12}{7}$$