x^2-4x-5=-x^2+2x+3 eching

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

x uchun yechish

$$x=-1$$

$$x=4$$

Guruhlash orqali faktorlash qadamlari

x^{2} ni ikki tarafga qo’shing.

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

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

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

Ikkala tarafdan 2x ni ayirish.

$$2x^{2}-4x-5-2x=3$$

-6x ni olish uchun -4x va -2x ni birlashtirish.

$$2x^{2}-6x-5=3$$

Ikkala tarafdan 3 ni ayirish.

$$2x^{2}-6x-5-3=0$$

-8 olish uchun -5 dan 3 ni ayirish.

$$2x^{2}-6x-8=0$$

Ikki tarafini 2 ga bo‘ling.

$$x^{2}-3x-4=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx-4 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-3 ab=1\left(-4\right)=-4$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -4-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-4 2,-2$$

Har bir juftlik yigʻindisini hisoblang.

$$1-4=-3 2-2=0$$

Yechim – -3 yigʻindisini beruvchi juftlik.

$$a=-4 b=1$$

x^{2}-3x-4 ni \left(x^{2}-4x\right)+\left(x-4\right) sifatida qaytadan yozish.

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

x^{2}-4x ichida x ni ajrating.

$$x\left(x-4\right)+x-4$$

Distributiv funktsiyasidan foydalangan holda x-4 umumiy terminini chiqaring.

$$\left(x-4\right)\left(x+1\right)$$

Tenglamani yechish uchun x-4=0 va x+1=0 ni yeching.

$$x=4 x=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

x^{2} ni ikki tarafga qo’shing.

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

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

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

Ikkala tarafdan 2x ni ayirish.

$$2x^{2}-4x-5-2x=3$$

-6x ni olish uchun -4x va -2x ni birlashtirish.

$$2x^{2}-6x-5=3$$

Ikkala tarafdan 3 ni ayirish.

$$2x^{2}-6x-5-3=0$$

-8 olish uchun -5 dan 3 ni ayirish.

$$2x^{2}-6x-8=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -6 ni b va -8 ni c bilan almashtiring.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\left(-8\right)}}{2\times 2}$$

-6 kvadratini chiqarish.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 2\left(-8\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36-8\left(-8\right)}}{2\times 2}$$

-8 ni -8 marotabaga ko'paytirish.

$$x=\frac{-\left(-6\right)±\sqrt{36+64}}{2\times 2}$$

36 ni 64 ga qo'shish.

$$x=\frac{-\left(-6\right)±\sqrt{100}}{2\times 2}$$

100 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-6\right)±10}{2\times 2}$$

-6 ning teskarisi 6 ga teng.

$$x=\frac{6±10}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{6±10}{4}$$

x=\frac{6±10}{4} tenglamasini yeching, bunda ± musbat. 6 ni 10 ga qo'shish.

$$x=\frac{16}{4}$$

16 ni 4 ga bo'lish.

$$x=4$$

x=\frac{6±10}{4} tenglamasini yeching, bunda ± manfiy. 6 dan 10 ni ayirish.

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

-4 ni 4 ga bo'lish.

$$x=-1$$

Tenglama yechildi.

$$x=4 x=-1$$

Kvadratni yakunlash bo'yicha qadamlar

x^{2} ni ikki tarafga qo’shing.

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

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

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

Ikkala tarafdan 2x ni ayirish.

$$2x^{2}-4x-5-2x=3$$

-6x ni olish uchun -4x va -2x ni birlashtirish.

$$2x^{2}-6x-5=3$$

5 ni ikki tarafga qo’shing.

$$2x^{2}-6x=3+5$$

8 olish uchun 3 va 5'ni qo'shing.

$$2x^{2}-6x=8$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-6x}{2}=\frac{8}{2}$$

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

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

-6 ni 2 ga bo'lish.

$$x^{2}-3x=\frac{8}{2}$$

8 ni 2 ga bo'lish.

$$x^{2}-3x=4$$

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$x^{2}-3x+\frac{9}{4}=4+\frac{9}{4}$$

4 ni \frac{9}{4} ga qo'shish.

$$x^{2}-3x+\frac{9}{4}=\frac{25}{4}$$

x^{2}-3x+\frac{9}{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.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{25}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Qisqartirish.

$$x-\frac{3}{2}=\frac{5}{2} x-\frac{3}{2}=-\frac{5}{2}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=4 x=-1$$