x^{2}+(x^{2}-2x)^2=10+(x+1)^2+(x^2-2x-3)^2 eching

$$x ^ { 2 } + ( x ^ { 2 } - 2 x ) ^ { 2 } = 10 + ( x + 1 ) ^ { 2 } + ( x ^ { 2 } - 2 x - 3 ) ^ { 2 }$$

x uchun yechish

$$x=-1$$

$$x = \frac{10}{3} = 3\frac{1}{3} \approx 3,333333333$$

Guruhlash orqali faktorlash qadamlari

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

$$x^{2}+\left(x^{2}\right)^{2}-4x^{2}x+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 2 ni ko‘paytirib, 4 ni oling.

$$x^{2}+x^{4}-4x^{2}x+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}$$

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 2 va 1 ni qo‘shib, 3 ni oling.

$$x^{2}+x^{4}-4x^{3}+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}$$

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

$$5x^{2}+x^{4}-4x^{3}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

$$5x^{2}+x^{4}-4x^{3}=10+x^{2}+2x+1+\left(x^{2}-2x-3\right)^{2}$$

11 olish uchun 10 va 1'ni qo'shing.

$$5x^{2}+x^{4}-4x^{3}=11+x^{2}+2x+\left(x^{2}-2x-3\right)^{2}$$

x^{2}-2x-3 kvadratini chiqarish.

$$5x^{2}+x^{4}-4x^{3}=11+x^{2}+2x+x^{4}-4x^{3}-2x^{2}+12x+9$$

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

$$5x^{2}+x^{4}-4x^{3}=11-x^{2}+2x+x^{4}-4x^{3}+12x+9$$

14x ni olish uchun 2x va 12x ni birlashtirish.

$$5x^{2}+x^{4}-4x^{3}=11-x^{2}+14x+x^{4}-4x^{3}+9$$

20 olish uchun 11 va 9'ni qo'shing.

$$5x^{2}+x^{4}-4x^{3}=20-x^{2}+14x+x^{4}-4x^{3}$$

Ikkala tarafdan 20 ni ayirish.

$$5x^{2}+x^{4}-4x^{3}-20=-x^{2}+14x+x^{4}-4x^{3}$$

x^{2} ni ikki tarafga qo’shing.

$$5x^{2}+x^{4}-4x^{3}-20+x^{2}=14x+x^{4}-4x^{3}$$

6x^{2} ni olish uchun 5x^{2} va x^{2} ni birlashtirish.

$$6x^{2}+x^{4}-4x^{3}-20=14x+x^{4}-4x^{3}$$

Ikkala tarafdan 14x ni ayirish.

$$6x^{2}+x^{4}-4x^{3}-20-14x=x^{4}-4x^{3}$$

Ikkala tarafdan x^{4} ni ayirish.

$$6x^{2}+x^{4}-4x^{3}-20-14x-x^{4}=-4x^{3}$$

0 ni olish uchun x^{4} va -x^{4} ni birlashtirish.

$$6x^{2}-4x^{3}-20-14x=-4x^{3}$$

4x^{3} ni ikki tarafga qo’shing.

$$6x^{2}-4x^{3}-20-14x+4x^{3}=0$$

0 ni olish uchun -4x^{3} va 4x^{3} ni birlashtirish.

$$6x^{2}-20-14x=0$$

Ikki tarafini 2 ga bo‘ling.

$$3x^{2}-10-7x=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$3x^{2}-7x-10=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 3x^{2}+ax+bx-10 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-7 ab=3\left(-10\right)=-30$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -30-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-30 2,-15 3,-10 5,-6$$

Har bir juftlik yigʻindisini hisoblang.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

Yechim – -7 yigʻindisini beruvchi juftlik.

$$a=-10 b=3$$

3x^{2}-7x-10 ni \left(3x^{2}-10x\right)+\left(3x-10\right) sifatida qaytadan yozish.

$$\left(3x^{2}-10x\right)+\left(3x-10\right)$$

3x^{2}-10x ichida x ni ajrating.

$$x\left(3x-10\right)+3x-10$$

Distributiv funktsiyasidan foydalangan holda 3x-10 umumiy terminini chiqaring.

$$\left(3x-10\right)\left(x+1\right)$$

Tenglamani yechish uchun 3x-10=0 va x+1=0 ni yeching.

$$x=\frac{10}{3} x=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$x^{2}+\left(x^{2}\right)^{2}-4x^{2}x+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 2 ni ko‘paytirib, 4 ni oling.

$$x^{2}+x^{4}-4x^{2}x+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}$$

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 2 va 1 ni qo‘shib, 3 ni oling.

$$x^{2}+x^{4}-4x^{3}+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}$$

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

$$5x^{2}+x^{4}-4x^{3}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

$$5x^{2}+x^{4}-4x^{3}=10+x^{2}+2x+1+\left(x^{2}-2x-3\right)^{2}$$

11 olish uchun 10 va 1'ni qo'shing.

$$5x^{2}+x^{4}-4x^{3}=11+x^{2}+2x+\left(x^{2}-2x-3\right)^{2}$$

x^{2}-2x-3 kvadratini chiqarish.

$$5x^{2}+x^{4}-4x^{3}=11+x^{2}+2x+x^{4}-4x^{3}-2x^{2}+12x+9$$

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

$$5x^{2}+x^{4}-4x^{3}=11-x^{2}+2x+x^{4}-4x^{3}+12x+9$$

14x ni olish uchun 2x va 12x ni birlashtirish.

$$5x^{2}+x^{4}-4x^{3}=11-x^{2}+14x+x^{4}-4x^{3}+9$$

20 olish uchun 11 va 9'ni qo'shing.

$$5x^{2}+x^{4}-4x^{3}=20-x^{2}+14x+x^{4}-4x^{3}$$

Ikkala tarafdan 20 ni ayirish.

$$5x^{2}+x^{4}-4x^{3}-20=-x^{2}+14x+x^{4}-4x^{3}$$

x^{2} ni ikki tarafga qo’shing.

$$5x^{2}+x^{4}-4x^{3}-20+x^{2}=14x+x^{4}-4x^{3}$$

6x^{2} ni olish uchun 5x^{2} va x^{2} ni birlashtirish.

$$6x^{2}+x^{4}-4x^{3}-20=14x+x^{4}-4x^{3}$$

Ikkala tarafdan 14x ni ayirish.

$$6x^{2}+x^{4}-4x^{3}-20-14x=x^{4}-4x^{3}$$

Ikkala tarafdan x^{4} ni ayirish.

$$6x^{2}+x^{4}-4x^{3}-20-14x-x^{4}=-4x^{3}$$

0 ni olish uchun x^{4} va -x^{4} ni birlashtirish.

$$6x^{2}-4x^{3}-20-14x=-4x^{3}$$

4x^{3} ni ikki tarafga qo’shing.

$$6x^{2}-4x^{3}-20-14x+4x^{3}=0$$

0 ni olish uchun -4x^{3} va 4x^{3} ni birlashtirish.

$$6x^{2}-20-14x=0$$

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.

$$6x^{2}-14x-20=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, -14 ni b va -20 ni c bilan almashtiring.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 6\left(-20\right)}}{2\times 6}$$

-14 kvadratini chiqarish.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 6\left(-20\right)}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-24\left(-20\right)}}{2\times 6}$$

-24 ni -20 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196+480}}{2\times 6}$$

196 ni 480 ga qo'shish.

$$x=\frac{-\left(-14\right)±\sqrt{676}}{2\times 6}$$

676 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-14\right)±26}{2\times 6}$$

-14 ning teskarisi 14 ga teng.

$$x=\frac{14±26}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$x=\frac{14±26}{12}$$

x=\frac{14±26}{12} tenglamasini yeching, bunda ± musbat. 14 ni 26 ga qo'shish.

$$x=\frac{40}{12}$$

\frac{40}{12} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{10}{3}$$

x=\frac{14±26}{12} tenglamasini yeching, bunda ± manfiy. 14 dan 26 ni ayirish.

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

-12 ni 12 ga bo'lish.

$$x=-1$$

Tenglama yechildi.

$$x=\frac{10}{3} x=-1$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$x^{2}+\left(x^{2}\right)^{2}-4x^{2}x+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 2 ni ko‘paytirib, 4 ni oling.

$$x^{2}+x^{4}-4x^{2}x+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}$$

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 2 va 1 ni qo‘shib, 3 ni oling.

$$x^{2}+x^{4}-4x^{3}+4x^{2}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}$$

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

$$5x^{2}+x^{4}-4x^{3}=10+\left(x+1\right)^{2}+\left(x^{2}-2x-3\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+1\right)^{2} kengaytirilishi uchun ishlating.

$$5x^{2}+x^{4}-4x^{3}=10+x^{2}+2x+1+\left(x^{2}-2x-3\right)^{2}$$

11 olish uchun 10 va 1'ni qo'shing.

$$5x^{2}+x^{4}-4x^{3}=11+x^{2}+2x+\left(x^{2}-2x-3\right)^{2}$$

x^{2}-2x-3 kvadratini chiqarish.

$$5x^{2}+x^{4}-4x^{3}=11+x^{2}+2x+x^{4}-4x^{3}-2x^{2}+12x+9$$

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

$$5x^{2}+x^{4}-4x^{3}=11-x^{2}+2x+x^{4}-4x^{3}+12x+9$$

14x ni olish uchun 2x va 12x ni birlashtirish.

$$5x^{2}+x^{4}-4x^{3}=11-x^{2}+14x+x^{4}-4x^{3}+9$$

20 olish uchun 11 va 9'ni qo'shing.

$$5x^{2}+x^{4}-4x^{3}=20-x^{2}+14x+x^{4}-4x^{3}$$

x^{2} ni ikki tarafga qo’shing.

$$5x^{2}+x^{4}-4x^{3}+x^{2}=20+14x+x^{4}-4x^{3}$$

6x^{2} ni olish uchun 5x^{2} va x^{2} ni birlashtirish.

$$6x^{2}+x^{4}-4x^{3}=20+14x+x^{4}-4x^{3}$$

Ikkala tarafdan 14x ni ayirish.

$$6x^{2}+x^{4}-4x^{3}-14x=20+x^{4}-4x^{3}$$

Ikkala tarafdan x^{4} ni ayirish.

$$6x^{2}+x^{4}-4x^{3}-14x-x^{4}=20-4x^{3}$$

0 ni olish uchun x^{4} va -x^{4} ni birlashtirish.

$$6x^{2}-4x^{3}-14x=20-4x^{3}$$

4x^{3} ni ikki tarafga qo’shing.

$$6x^{2}-4x^{3}-14x+4x^{3}=20$$

0 ni olish uchun -4x^{3} va 4x^{3} ni birlashtirish.

$$6x^{2}-14x=20$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6x^{2}-14x}{6}=\frac{20}{6}$$

6 ga bo'lish 6 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{14}{6}\right)x=\frac{20}{6}$$

\frac{-14}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{7}{3}x=\frac{20}{6}$$

\frac{20}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{7}{3}x=\frac{10}{3}$$

-\frac{7}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{6} olish uchun. Keyin, -\frac{7}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=\frac{10}{3}+\left(-\frac{7}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{6} kvadratini chiqarish.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{10}{3}+\frac{49}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{10}{3} ni \frac{49}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{169}{36}$$

x^{2}-\frac{7}{3}x+\frac{49}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{7}{6}\right)^{2}=\frac{169}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{\frac{169}{36}}$$

Qisqartirish.

$$x-\frac{7}{6}=\frac{13}{6} x-\frac{7}{6}=-\frac{13}{6}$$

\frac{7}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{10}{3} x=-1$$