-x^2+14x-13=2x^2+6x-18-6x eching

$$- x ^ { 2 } + 14 x - 13 = 2 x ^ { 2 } + 6 x - 18 - 6 x$$

x uchun yechish

$$x=5$$

$$x=-\frac{1}{3}\approx -0,333333333$$

Guruhlash orqali faktorlash qadamlari

0 ni olish uchun 6x va -6x ni birlashtirish.

$$-x^{2}+14x-13=2x^{2}-18$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$-x^{2}+14x-13-2x^{2}=-18$$

18 ni ikki tarafga qo’shing.

$$-x^{2}+14x-13-2x^{2}+18=0$$

5 olish uchun -13 va 18'ni qo'shing.

$$-x^{2}+14x+5-2x^{2}=0$$

-3x^{2} ni olish uchun -x^{2} va -2x^{2} ni birlashtirish.

$$-3x^{2}+14x+5=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -3x^{2}+ax+bx+5 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=14 ab=-3\times 5=-15$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -15-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,15 -3,5$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+15=14 -3+5=2$$

Yechim – 14 yigʻindisini beruvchi juftlik.

$$a=15 b=-1$$

-3x^{2}+14x+5 ni \left(-3x^{2}+15x\right)+\left(-x+5\right) sifatida qaytadan yozish.

$$\left(-3x^{2}+15x\right)+\left(-x+5\right)$$

-3x^{2}+15x ichida 3x ni ajrating.

$$3x\left(-x+5\right)-x+5$$

Distributiv funktsiyasidan foydalangan holda -x+5 umumiy terminini chiqaring.

$$\left(-x+5\right)\left(3x+1\right)$$

Tenglamani yechish uchun -x+5=0 va 3x+1=0 ni yeching.

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

Kvadrat tenglamalaridan foydalanish qoidalari

0 ni olish uchun 6x va -6x ni birlashtirish.

$$-x^{2}+14x-13=2x^{2}-18$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$-x^{2}+14x-13-2x^{2}=-18$$

18 ni ikki tarafga qo’shing.

$$-x^{2}+14x-13-2x^{2}+18=0$$

5 olish uchun -13 va 18'ni qo'shing.

$$-x^{2}+14x+5-2x^{2}=0$$

-3x^{2} ni olish uchun -x^{2} va -2x^{2} ni birlashtirish.

$$-3x^{2}+14x+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, 14 ni b va 5 ni c bilan almashtiring.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

14 kvadratini chiqarish.

$$x=\frac{-14±\sqrt{196-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-14±\sqrt{196+12\times 5}}{2\left(-3\right)}$$

12 ni 5 marotabaga ko'paytirish.

$$x=\frac{-14±\sqrt{196+60}}{2\left(-3\right)}$$

196 ni 60 ga qo'shish.

$$x=\frac{-14±\sqrt{256}}{2\left(-3\right)}$$

256 ning kvadrat ildizini chiqarish.

$$x=\frac{-14±16}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{-14±16}{-6}$$

x=\frac{-14±16}{-6} tenglamasini yeching, bunda ± musbat. -14 ni 16 ga qo'shish.

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

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

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

x=\frac{-14±16}{-6} tenglamasini yeching, bunda ± manfiy. -14 dan 16 ni ayirish.

$$x=-\frac{30}{-6}$$

-30 ni -6 ga bo'lish.

$$x=5$$

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

0 ni olish uchun 6x va -6x ni birlashtirish.

$$-x^{2}+14x-13=2x^{2}-18$$

Ikkala tarafdan 2x^{2} ni ayirish.

$$-x^{2}+14x-13-2x^{2}=-18$$

13 ni ikki tarafga qo’shing.

$$-x^{2}+14x-2x^{2}=-18+13$$

-5 olish uchun -18 va 13'ni qo'shing.

$$-x^{2}+14x-2x^{2}=-5$$

-3x^{2} ni olish uchun -x^{2} va -2x^{2} ni birlashtirish.

$$-3x^{2}+14x=-5$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}+14x}{-3}=-\frac{5}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{14}{-3}x=-\frac{5}{-3}$$

14 ni -3 ga bo'lish.

$$x^{2}-\frac{14}{3}x=-\frac{5}{-3}$$

-5 ni -3 ga bo'lish.

$$x^{2}-\frac{14}{3}x=\frac{5}{3}$$

-\frac{14}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{3} olish uchun. Keyin, -\frac{7}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{14}{3}x+\left(-\frac{7}{3}\right)^{2}=\frac{5}{3}+\left(-\frac{7}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{3} kvadratini chiqarish.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=\frac{5}{3}+\frac{49}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{3} ni \frac{49}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=\frac{64}{9}$$

x^{2}-\frac{14}{3}x+\frac{49}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{7}{3}\right)^{2}=\frac{64}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{3}\right)^{2}}=\sqrt{\frac{64}{9}}$$

Qisqartirish.

$$x-\frac{7}{3}=\frac{8}{3} x-\frac{7}{3}=-\frac{8}{3}$$

\frac{7}{3} ni tenglamaning ikkala tarafiga qo'shish.

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