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

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

x uchun yechish

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

$$x=1$$

Guruhlash orqali faktorlash qadamlari

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

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

3x ni olish uchun 1x va 2x ni birlashtirish.

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

Ikkala tarafdan 5 ni ayirish.

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

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

$$a+b=3 ab=2\left(-5\right)=-10$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -10-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,10 -2,5$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+10=9 -2+5=3$$

Yechim – 3 yigʻindisini beruvchi juftlik.

$$a=-2 b=5$$

2x^{2}+3x-5 ni \left(2x^{2}-2x\right)+\left(5x-5\right) sifatida qaytadan yozish.

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

Birinchi guruhda 2x ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$2x\left(x-1\right)+5\left(x-1\right)$$

Distributiv funktsiyasidan foydalangan holda x-1 umumiy terminini chiqaring.

$$\left(x-1\right)\left(2x+5\right)$$

Tenglamani yechish uchun x-1=0 va 2x+5=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

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

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

3x ni olish uchun 1x va 2x ni birlashtirish.

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

Ikkala tarafdan 5 ni ayirish.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 3 ni b va -5 ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}-4\times 2\left(-5\right)}}{2\times 2}$$

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\times 2\left(-5\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9-8\left(-5\right)}}{2\times 2}$$

-8 ni -5 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+40}}{2\times 2}$$

9 ni 40 ga qo'shish.

$$x=\frac{-3±\sqrt{49}}{2\times 2}$$

49 ning kvadrat ildizini chiqarish.

$$x=\frac{-3±7}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-3±7}{4}$$

x=\frac{-3±7}{4} tenglamasini yeching, bunda ± musbat. -3 ni 7 ga qo'shish.

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

4 ni 4 ga bo'lish.

$$x=1$$

x=\frac{-3±7}{4} tenglamasini yeching, bunda ± manfiy. -3 dan 7 ni ayirish.

$$x=-\frac{10}{4}$$

\frac{-10}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

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

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

3x ni olish uchun 1x va 2x ni birlashtirish.

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

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+3x}{2}=\frac{5}{2}$$

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

$$x^{2}+\frac{3}{2}x=\frac{5}{2}$$

\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{4} olish uchun. Keyin, \frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=\frac{5}{2}+\left(\frac{3}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{4} kvadratini chiqarish.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{5}{2}+\frac{9}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{2} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{49}{16}$$

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

$$\left(x+\frac{3}{4}\right)^{2}=\frac{49}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Qisqartirish.

$$x+\frac{3}{4}=\frac{7}{4} x+\frac{3}{4}=-\frac{7}{4}$$

Tenglamaning ikkala tarafidan \frac{3}{4} ni ayirish.

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