4x^2+3x+1=3-4x eching

$$4 x ^ { 2 } + 3 x + 1 = 3 - 4 x$$

x uchun yechish

$$x=-2$$

$$x=\frac{1}{4}=0,25$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 3 ni ayirish.

$$4x^{2}+3x+1-3=-4x$$

-2 olish uchun 1 dan 3 ni ayirish.

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

4x ni ikki tarafga qo’shing.

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

7x ni olish uchun 3x va 4x ni birlashtirish.

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

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

$$a+b=7 ab=4\left(-2\right)=-8$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -8-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,8 -2,4$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+8=7 -2+4=2$$

Yechim – 7 yigʻindisini beruvchi juftlik.

$$a=-1 b=8$$

4x^{2}+7x-2 ni \left(4x^{2}-x\right)+\left(8x-2\right) sifatida qaytadan yozish.

$$\left(4x^{2}-x\right)+\left(8x-2\right)$$

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

$$x\left(4x-1\right)+2\left(4x-1\right)$$

Distributiv funktsiyasidan foydalangan holda 4x-1 umumiy terminini chiqaring.

$$\left(4x-1\right)\left(x+2\right)$$

Tenglamani yechish uchun 4x-1=0 va x+2=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 3 ni ayirish.

$$4x^{2}+3x+1-3=-4x$$

-2 olish uchun 1 dan 3 ni ayirish.

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

4x ni ikki tarafga qo’shing.

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

7x ni olish uchun 3x va 4x ni birlashtirish.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, 7 ni b va -2 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\times 4\left(-2\right)}}{2\times 4}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\times 4\left(-2\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-16\left(-2\right)}}{2\times 4}$$

-16 ni -2 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49+32}}{2\times 4}$$

49 ni 32 ga qo'shish.

$$x=\frac{-7±\sqrt{81}}{2\times 4}$$

81 ning kvadrat ildizini chiqarish.

$$x=\frac{-7±9}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{-7±9}{8}$$

x=\frac{-7±9}{8} tenglamasini yeching, bunda ± musbat. -7 ni 9 ga qo'shish.

$$x=\frac{2}{8}$$

\frac{2}{8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{-7±9}{8} tenglamasini yeching, bunda ± manfiy. -7 dan 9 ni ayirish.

$$x=-\frac{16}{8}$$

-16 ni 8 ga bo'lish.

$$x=-2$$

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

4x ni ikki tarafga qo’shing.

$$4x^{2}+3x+1+4x=3$$

7x ni olish uchun 3x va 4x ni birlashtirish.

$$4x^{2}+7x+1=3$$

Ikkala tarafdan 1 ni ayirish.

$$4x^{2}+7x=3-1$$

2 olish uchun 3 dan 1 ni ayirish.

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

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}+7x}{4}=\frac{2}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{7}{4}x=\frac{2}{4}$$

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

$$x^{2}+\frac{7}{4}x=\frac{1}{2}$$

\frac{7}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{8} olish uchun. Keyin, \frac{7}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{7}{4}x+\left(\frac{7}{8}\right)^{2}=\frac{1}{2}+\left(\frac{7}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{8} kvadratini chiqarish.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{1}{2}+\frac{49}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{49}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{81}{64}$$

x^{2}+\frac{7}{4}x+\frac{49}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{7}{8}\right)^{2}=\frac{81}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{8}\right)^{2}}=\sqrt{\frac{81}{64}}$$

Qisqartirish.

$$x+\frac{7}{8}=\frac{9}{8} x+\frac{7}{8}=-\frac{9}{8}$$

Tenglamaning ikkala tarafidan \frac{7}{8} ni ayirish.

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