(2x+3)^2-15^2=10^2-(x-1)^2 eching

$$( 2 x + 3 ) ^ { 2 } - 15 ^ { 2 } = 10 ^ { 2 } - ( x - 1 ) ^ { 2 }$$

x uchun yechish

$$x=-9$$

$$x=7$$

Guruhlash orqali faktorlash qadamlari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+3\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}+12x+9-15^{2}=10^{2}-\left(x-1\right)^{2}$$

2 daraja ko‘rsatkichini 15 ga hisoblang va 225 ni qiymatni oling.

$$4x^{2}+12x+9-225=10^{2}-\left(x-1\right)^{2}$$

-216 olish uchun 9 dan 225 ni ayirish.

$$4x^{2}+12x-216=10^{2}-\left(x-1\right)^{2}$$

2 daraja ko‘rsatkichini 10 ga hisoblang va 100 ni qiymatni oling.

$$4x^{2}+12x-216=100-\left(x-1\right)^{2}$$

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

$$4x^{2}+12x-216=100-\left(x^{2}-2x+1\right)$$

x^{2}-2x+1 teskarisini topish uchun har birining teskarisini toping.

$$4x^{2}+12x-216=100-x^{2}+2x-1$$

99 olish uchun 100 dan 1 ni ayirish.

$$4x^{2}+12x-216=99-x^{2}+2x$$

Ikkala tarafdan 99 ni ayirish.

$$4x^{2}+12x-216-99=-x^{2}+2x$$

-315 olish uchun -216 dan 99 ni ayirish.

$$4x^{2}+12x-315=-x^{2}+2x$$

x^{2} ni ikki tarafga qo’shing.

$$4x^{2}+12x-315+x^{2}=2x$$

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

$$5x^{2}+12x-315=2x$$

Ikkala tarafdan 2x ni ayirish.

$$5x^{2}+12x-315-2x=0$$

10x ni olish uchun 12x va -2x ni birlashtirish.

$$5x^{2}+10x-315=0$$

Ikki tarafini 5 ga bo‘ling.

$$x^{2}+2x-63=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx-63 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=2 ab=1\left(-63\right)=-63$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -63-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,63 -3,21 -7,9$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+63=62 -3+21=18 -7+9=2$$

Yechim – 2 yigʻindisini beruvchi juftlik.

$$a=-7 b=9$$

x^{2}+2x-63 ni \left(x^{2}-7x\right)+\left(9x-63\right) sifatida qaytadan yozish.

$$\left(x^{2}-7x\right)+\left(9x-63\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 9 ni faktordan chiqaring.

$$x\left(x-7\right)+9\left(x-7\right)$$

Distributiv funktsiyasidan foydalangan holda x-7 umumiy terminini chiqaring.

$$\left(x-7\right)\left(x+9\right)$$

Tenglamani yechish uchun x-7=0 va x+9=0 ni yeching.

$$x=7 x=-9$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+3\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}+12x+9-15^{2}=10^{2}-\left(x-1\right)^{2}$$

2 daraja ko‘rsatkichini 15 ga hisoblang va 225 ni qiymatni oling.

$$4x^{2}+12x+9-225=10^{2}-\left(x-1\right)^{2}$$

-216 olish uchun 9 dan 225 ni ayirish.

$$4x^{2}+12x-216=10^{2}-\left(x-1\right)^{2}$$

2 daraja ko‘rsatkichini 10 ga hisoblang va 100 ni qiymatni oling.

$$4x^{2}+12x-216=100-\left(x-1\right)^{2}$$

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

$$4x^{2}+12x-216=100-\left(x^{2}-2x+1\right)$$

x^{2}-2x+1 teskarisini topish uchun har birining teskarisini toping.

$$4x^{2}+12x-216=100-x^{2}+2x-1$$

99 olish uchun 100 dan 1 ni ayirish.

$$4x^{2}+12x-216=99-x^{2}+2x$$

Ikkala tarafdan 99 ni ayirish.

$$4x^{2}+12x-216-99=-x^{2}+2x$$

-315 olish uchun -216 dan 99 ni ayirish.

$$4x^{2}+12x-315=-x^{2}+2x$$

x^{2} ni ikki tarafga qo’shing.

$$4x^{2}+12x-315+x^{2}=2x$$

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

$$5x^{2}+12x-315=2x$$

Ikkala tarafdan 2x ni ayirish.

$$5x^{2}+12x-315-2x=0$$

10x ni olish uchun 12x va -2x ni birlashtirish.

$$5x^{2}+10x-315=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, 10 ni b va -315 ni c bilan almashtiring.

$$x=\frac{-10±\sqrt{10^{2}-4\times 5\left(-315\right)}}{2\times 5}$$

10 kvadratini chiqarish.

$$x=\frac{-10±\sqrt{100-4\times 5\left(-315\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-10±\sqrt{100-20\left(-315\right)}}{2\times 5}$$

-20 ni -315 marotabaga ko'paytirish.

$$x=\frac{-10±\sqrt{100+6300}}{2\times 5}$$

100 ni 6300 ga qo'shish.

$$x=\frac{-10±\sqrt{6400}}{2\times 5}$$

6400 ning kvadrat ildizini chiqarish.

$$x=\frac{-10±80}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{-10±80}{10}$$

x=\frac{-10±80}{10} tenglamasini yeching, bunda ± musbat. -10 ni 80 ga qo'shish.

$$x=\frac{70}{10}$$

70 ni 10 ga bo'lish.

$$x=7$$

x=\frac{-10±80}{10} tenglamasini yeching, bunda ± manfiy. -10 dan 80 ni ayirish.

$$x=-\frac{90}{10}$$

-90 ni 10 ga bo'lish.

$$x=-9$$

Tenglama yechildi.

$$x=7 x=-9$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+3\right)^{2} kengaytirilishi uchun ishlating.

$$4x^{2}+12x+9-15^{2}=10^{2}-\left(x-1\right)^{2}$$

2 daraja ko‘rsatkichini 15 ga hisoblang va 225 ni qiymatni oling.

$$4x^{2}+12x+9-225=10^{2}-\left(x-1\right)^{2}$$

-216 olish uchun 9 dan 225 ni ayirish.

$$4x^{2}+12x-216=10^{2}-\left(x-1\right)^{2}$$

2 daraja ko‘rsatkichini 10 ga hisoblang va 100 ni qiymatni oling.

$$4x^{2}+12x-216=100-\left(x-1\right)^{2}$$

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

$$4x^{2}+12x-216=100-\left(x^{2}-2x+1\right)$$

x^{2}-2x+1 teskarisini topish uchun har birining teskarisini toping.

$$4x^{2}+12x-216=100-x^{2}+2x-1$$

99 olish uchun 100 dan 1 ni ayirish.

$$4x^{2}+12x-216=99-x^{2}+2x$$

x^{2} ni ikki tarafga qo’shing.

$$4x^{2}+12x-216+x^{2}=99+2x$$

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

$$5x^{2}+12x-216=99+2x$$

Ikkala tarafdan 2x ni ayirish.

$$5x^{2}+12x-216-2x=99$$

10x ni olish uchun 12x va -2x ni birlashtirish.

$$5x^{2}+10x-216=99$$

216 ni ikki tarafga qo’shing.

$$5x^{2}+10x=99+216$$

315 olish uchun 99 va 216'ni qo'shing.

$$5x^{2}+10x=315$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}+10x}{5}=\frac{315}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{10}{5}x=\frac{315}{5}$$

10 ni 5 ga bo'lish.

$$x^{2}+2x=\frac{315}{5}$$

315 ni 5 ga bo'lish.

$$x^{2}+2x=63$$

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.

$$x^{2}+2x+1^{2}=63+1^{2}$$

1 kvadratini chiqarish.

$$x^{2}+2x+1=63+1$$

63 ni 1 ga qo'shish.

$$x^{2}+2x+1=64$$

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.

$$\left(x+1\right)^{2}=64$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{64}$$

Qisqartirish.

$$x+1=8 x+1=-8$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$x=7 x=-9$$