-8x^2=-14x-15 eching

$$- 8 x ^ { 2 } = - 14 x - 15$$

x uchun yechish

$$x=-\frac{3}{4}=-0,75$$

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

Guruhlash orqali faktorlash qadamlari

14x ni ikki tarafga qo’shing.

$$-8x^{2}+14x=-15$$

15 ni ikki tarafga qo’shing.

$$-8x^{2}+14x+15=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -8x^{2}+ax+bx+15 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=14 ab=-8\times 15=-120$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -120-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,120 -2,60 -3,40 -4,30 -5,24 -6,20 -8,15 -10,12$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+120=119 -2+60=58 -3+40=37 -4+30=26 -5+24=19 -6+20=14 -8+15=7 -10+12=2$$

Yechim – 14 yigʻindisini beruvchi juftlik.

$$a=20 b=-6$$

-8x^{2}+14x+15 ni \left(-8x^{2}+20x\right)+\left(-6x+15\right) sifatida qaytadan yozish.

$$\left(-8x^{2}+20x\right)+\left(-6x+15\right)$$

Birinchi guruhda -4x ni va ikkinchi guruhda -3 ni faktordan chiqaring.

$$-4x\left(2x-5\right)-3\left(2x-5\right)$$

Distributiv funktsiyasidan foydalangan holda 2x-5 umumiy terminini chiqaring.

$$\left(2x-5\right)\left(-4x-3\right)$$

Tenglamani yechish uchun 2x-5=0 va -4x-3=0 ni yeching.

$$x=\frac{5}{2} x=-\frac{3}{4}$$

Kvadrat tenglamalaridan foydalanish qoidalari

14x ni ikki tarafga qo’shing.

$$-8x^{2}+14x=-15$$

15 ni ikki tarafga qo’shing.

$$-8x^{2}+14x+15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -8 ni a, 14 ni b va 15 ni c bilan almashtiring.

$$x=\frac{-14±\sqrt{14^{2}-4\left(-8\right)\times 15}}{2\left(-8\right)}$$

14 kvadratini chiqarish.

$$x=\frac{-14±\sqrt{196-4\left(-8\right)\times 15}}{2\left(-8\right)}$$

-4 ni -8 marotabaga ko'paytirish.

$$x=\frac{-14±\sqrt{196+32\times 15}}{2\left(-8\right)}$$

32 ni 15 marotabaga ko'paytirish.

$$x=\frac{-14±\sqrt{196+480}}{2\left(-8\right)}$$

196 ni 480 ga qo'shish.

$$x=\frac{-14±\sqrt{676}}{2\left(-8\right)}$$

676 ning kvadrat ildizini chiqarish.

$$x=\frac{-14±26}{2\left(-8\right)}$$

2 ni -8 marotabaga ko'paytirish.

$$x=\frac{-14±26}{-16}$$

x=\frac{-14±26}{-16} tenglamasini yeching, bunda ± musbat. -14 ni 26 ga qo'shish.

$$x=\frac{12}{-16}$$

\frac{12}{-16} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{-14±26}{-16} tenglamasini yeching, bunda ± manfiy. -14 dan 26 ni ayirish.

$$x=-\frac{40}{-16}$$

\frac{-40}{-16} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

$$x=-\frac{3}{4} x=\frac{5}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

14x ni ikki tarafga qo’shing.

$$-8x^{2}+14x=-15$$

Ikki tarafini -8 ga bo‘ling.

$$\frac{-8x^{2}+14x}{-8}=-\frac{15}{-8}$$

-8 ga bo'lish -8 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{14}{-8}x=-\frac{15}{-8}$$

\frac{14}{-8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{7}{4}x=-\frac{15}{-8}$$

-15 ni -8 ga bo'lish.

$$x^{2}-\frac{7}{4}x=\frac{15}{8}$$

-\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.

$$x^{2}-\frac{7}{4}x+\left(-\frac{7}{8}\right)^{2}=\frac{15}{8}+\left(-\frac{7}{8}\right)^{2}$$

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

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=\frac{15}{8}+\frac{49}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{15}{8} ni \frac{49}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=\frac{169}{64}$$

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.

$$\left(x-\frac{7}{8}\right)^{2}=\frac{169}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{8}\right)^{2}}=\sqrt{\frac{169}{64}}$$

Qisqartirish.

$$x-\frac{7}{8}=\frac{13}{8} x-\frac{7}{8}=-\frac{13}{8}$$

\frac{7}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5}{2} x=-\frac{3}{4}$$