2x^2+x-5=2x+1 eching

$$2 x ^ { 2 } + x - 5 = 2 x + 1$$

x uchun yechish

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1,5$$

$$x=2$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 2x ni ayirish.

$$2x^{2}+x-5-2x=1$$

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

$$2x^{2}-x-5=1$$

Ikkala tarafdan 1 ni ayirish.

$$2x^{2}-x-5-1=0$$

-6 olish uchun -5 dan 1 ni ayirish.

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

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2x^{2}+ax+bx-6 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-1 ab=2\left(-6\right)=-12$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -12-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-12 2,-6 3,-4$$

Har bir juftlik yigʻindisini hisoblang.

$$1-12=-11 2-6=-4 3-4=-1$$

Yechim – -1 yigʻindisini beruvchi juftlik.

$$a=-4 b=3$$

2x^{2}-x-6 ni \left(2x^{2}-4x\right)+\left(3x-6\right) sifatida qaytadan yozish.

$$\left(2x^{2}-4x\right)+\left(3x-6\right)$$

Birinchi guruhda 2x ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$2x\left(x-2\right)+3\left(x-2\right)$$

Distributiv funktsiyasidan foydalangan holda x-2 umumiy terminini chiqaring.

$$\left(x-2\right)\left(2x+3\right)$$

Tenglamani yechish uchun x-2=0 va 2x+3=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 2x ni ayirish.

$$2x^{2}+x-5-2x=1$$

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

$$2x^{2}-x-5=1$$

Ikkala tarafdan 1 ni ayirish.

$$2x^{2}-x-5-1=0$$

-6 olish uchun -5 dan 1 ni ayirish.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -1 ni b va -6 ni c bilan almashtiring.

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

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-8\left(-6\right)}}{2\times 2}$$

-8 ni -6 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+48}}{2\times 2}$$

1 ni 48 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{49}}{2\times 2}$$

49 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1\right)±7}{2\times 2}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±7}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{1±7}{4}$$

x=\frac{1±7}{4} tenglamasini yeching, bunda ± musbat. 1 ni 7 ga qo'shish.

$$x=\frac{8}{4}$$

8 ni 4 ga bo'lish.

$$x=2$$

x=\frac{1±7}{4} tenglamasini yeching, bunda ± manfiy. 1 dan 7 ni ayirish.

$$x=-\frac{6}{4}$$

\frac{-6}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 2x ni ayirish.

$$2x^{2}+x-5-2x=1$$

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

$$2x^{2}-x-5=1$$

5 ni ikki tarafga qo’shing.

$$2x^{2}-x=1+5$$

6 olish uchun 1 va 5'ni qo'shing.

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

Ikki tarafini 2 ga bo‘ling.

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

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

$$x^{2}-\frac{1}{2}x=\frac{6}{2}$$

6 ni 2 ga bo'lish.

$$x^{2}-\frac{1}{2}x=3$$

-\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{4} olish uchun. Keyin, -\frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{4} kvadratini chiqarish.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=3+\frac{1}{16}$$

3 ni \frac{1}{16} ga qo'shish.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{49}{16}$$

x^{2}-\frac{1}{2}x+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{49}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Qisqartirish.

$$x-\frac{1}{4}=\frac{7}{4} x-\frac{1}{4}=-\frac{7}{4}$$

\frac{1}{4} ni tenglamaning ikkala tarafiga qo'shish.

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