x^2-x+12=2x^2+3x+7 eching

$$x ^ { 2 } - x + 12 = 2 x ^ { 2 } + 3 x + 7$$

x uchun yechish

$$x=-5$$

$$x=1$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 2x^{2} ni ayirish.

$$x^{2}-x+12-2x^{2}=3x+7$$

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

$$-x^{2}-x+12=3x+7$$

Ikkala tarafdan 3x ni ayirish.

$$-x^{2}-x+12-3x=7$$

-4x ni olish uchun -x va -3x ni birlashtirish.

$$-x^{2}-4x+12=7$$

Ikkala tarafdan 7 ni ayirish.

$$-x^{2}-4x+12-7=0$$

5 olish uchun 12 dan 7 ni ayirish.

$$-x^{2}-4x+5=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx+5 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-4 ab=-5=-5$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. Faqat bundan juftlik tizim yechimidir.

$$a=1 b=-5$$

-x^{2}-4x+5 ni \left(-x^{2}+x\right)+\left(-5x+5\right) sifatida qaytadan yozish.

$$\left(-x^{2}+x\right)+\left(-5x+5\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$x\left(-x+1\right)+5\left(-x+1\right)$$

Distributiv funktsiyasidan foydalangan holda -x+1 umumiy terminini chiqaring.

$$\left(-x+1\right)\left(x+5\right)$$

Tenglamani yechish uchun -x+1=0 va x+5=0 ni yeching.

$$x=1 x=-5$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 2x^{2} ni ayirish.

$$x^{2}-x+12-2x^{2}=3x+7$$

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

$$-x^{2}-x+12=3x+7$$

Ikkala tarafdan 3x ni ayirish.

$$-x^{2}-x+12-3x=7$$

-4x ni olish uchun -x va -3x ni birlashtirish.

$$-x^{2}-4x+12=7$$

Ikkala tarafdan 7 ni ayirish.

$$-x^{2}-4x+12-7=0$$

5 olish uchun 12 dan 7 ni ayirish.

$$-x^{2}-4x+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -4 ni b va 5 ni c bilan almashtiring.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 5}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 5}}{2\left(-1\right)}$$

4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16+20}}{2\left(-1\right)}$$

16 ni 20 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{36}}{2\left(-1\right)}$$

36 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±6}{2\left(-1\right)}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±6}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{4±6}{-2}$$

x=\frac{4±6}{-2} tenglamasini yeching, bunda ± musbat. 4 ni 6 ga qo'shish.

$$x=\frac{10}{-2}$$

10 ni -2 ga bo'lish.

$$x=-5$$

x=\frac{4±6}{-2} tenglamasini yeching, bunda ± manfiy. 4 dan 6 ni ayirish.

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

-2 ni -2 ga bo'lish.

$$x=1$$

Tenglama yechildi.

$$x=-5 x=1$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 2x^{2} ni ayirish.

$$x^{2}-x+12-2x^{2}=3x+7$$

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

$$-x^{2}-x+12=3x+7$$

Ikkala tarafdan 3x ni ayirish.

$$-x^{2}-x+12-3x=7$$

-4x ni olish uchun -x va -3x ni birlashtirish.

$$-x^{2}-4x+12=7$$

Ikkala tarafdan 12 ni ayirish.

$$-x^{2}-4x=7-12$$

-5 olish uchun 7 dan 12 ni ayirish.

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

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-4x}{-1}=-\frac{5}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

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

-4 ni -1 ga bo'lish.

$$x^{2}+4x=-\frac{5}{-1}$$

-5 ni -1 ga bo'lish.

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

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

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

2 kvadratini chiqarish.

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

5 ni 4 ga qo'shish.

$$x^{2}+4x+4=9$$

x^{2}+4x+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.

$$\left(x+2\right)^{2}=9$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{9}$$

Qisqartirish.

$$x+2=3 x+2=-3$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$x=1 x=-5$$