4x^2+8x=4x+8 eching

$$4 { x }^{ 2 } +8x = 4x+8$$

x uchun yechish

$$x=-2$$

$$x=1$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 4x ni ayirish.

$$4x^{2}+8x-4x=8$$

4x ni olish uchun 8x va -4x ni birlashtirish.

$$4x^{2}+4x=8$$

Ikkala tarafdan 8 ni ayirish.

$$4x^{2}+4x-8=0$$

Ikki tarafini 4 ga bo‘ling.

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

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

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

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

$$a=-1 b=2$$

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

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

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

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

Distributiv funktsiyasidan foydalangan holda x-1 umumiy terminini chiqaring.

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

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

$$x=1 x=-2$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 4x ni ayirish.

$$4x^{2}+8x-4x=8$$

4x ni olish uchun 8x va -4x ni birlashtirish.

$$4x^{2}+4x=8$$

Ikkala tarafdan 8 ni ayirish.

$$4x^{2}+4x-8=0$$

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

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

4 kvadratini chiqarish.

$$x=\frac{-4±\sqrt{16-4\times 4\left(-8\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16-16\left(-8\right)}}{2\times 4}$$

-16 ni -8 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16+128}}{2\times 4}$$

16 ni 128 ga qo'shish.

$$x=\frac{-4±\sqrt{144}}{2\times 4}$$

144 ning kvadrat ildizini chiqarish.

$$x=\frac{-4±12}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{-4±12}{8}$$

x=\frac{-4±12}{8} tenglamasini yeching, bunda ± musbat. -4 ni 12 ga qo'shish.

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

8 ni 8 ga bo'lish.

$$x=1$$

x=\frac{-4±12}{8} tenglamasini yeching, bunda ± manfiy. -4 dan 12 ni ayirish.

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

-16 ni 8 ga bo'lish.

$$x=-2$$

Tenglama yechildi.

$$x=1 x=-2$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 4x ni ayirish.

$$4x^{2}+8x-4x=8$$

4x ni olish uchun 8x va -4x ni birlashtirish.

$$4x^{2}+4x=8$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}+4x}{4}=\frac{8}{4}$$

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

$$x^{2}+\frac{4}{4}x=\frac{8}{4}$$

4 ni 4 ga bo'lish.

$$x^{2}+x=\frac{8}{4}$$

8 ni 4 ga bo'lish.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

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

2 ni \frac{1}{4} ga qo'shish.

$$x^{2}+x+\frac{1}{4}=\frac{9}{4}$$

x^{2}+x+\frac{1}{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.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{9}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Qisqartirish.

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

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$x=1 x=-2$$