-2x^2+13x+24=0 eching

$$- 2 x ^ { 2 } + 13 x + 24 = 0$$

x uchun yechish

$$x=8$$

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1,5$$

Guruhlash orqali faktorlash qadamlari

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -2x^{2}+ax+bx+24 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=13 ab=-2\times 24=-48$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -48-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,48 -2,24 -3,16 -4,12 -6,8$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+48=47 -2+24=22 -3+16=13 -4+12=8 -6+8=2$$

Yechim – 13 yigʻindisini beruvchi juftlik.

$$a=16 b=-3$$

-2x^{2}+13x+24 ni \left(-2x^{2}+16x\right)+\left(-3x+24\right) sifatida qaytadan yozish.

$$\left(-2x^{2}+16x\right)+\left(-3x+24\right)$$

Birinchi guruhda 2x ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$2x\left(-x+8\right)+3\left(-x+8\right)$$

Distributiv funktsiyasidan foydalangan holda -x+8 umumiy terminini chiqaring.

$$\left(-x+8\right)\left(2x+3\right)$$

Tenglamani yechish uchun -x+8=0 va 2x+3=0 ni yeching.

$$x=8 x=-\frac{3}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-2x^{2}+13x+24=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2 ni a, 13 ni b va 24 ni c bilan almashtiring.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-2\right)\times 24}}{2\left(-2\right)}$$

13 kvadratini chiqarish.

$$x=\frac{-13±\sqrt{169-4\left(-2\right)\times 24}}{2\left(-2\right)}$$

-4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-13±\sqrt{169+8\times 24}}{2\left(-2\right)}$$

8 ni 24 marotabaga ko'paytirish.

$$x=\frac{-13±\sqrt{169+192}}{2\left(-2\right)}$$

169 ni 192 ga qo'shish.

$$x=\frac{-13±\sqrt{361}}{2\left(-2\right)}$$

361 ning kvadrat ildizini chiqarish.

$$x=\frac{-13±19}{2\left(-2\right)}$$

2 ni -2 marotabaga ko'paytirish.

$$x=\frac{-13±19}{-4}$$

x=\frac{-13±19}{-4} tenglamasini yeching, bunda ± musbat. -13 ni 19 ga qo'shish.

$$x=\frac{6}{-4}$$

\frac{6}{-4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{-13±19}{-4} tenglamasini yeching, bunda ± manfiy. -13 dan 19 ni ayirish.

$$x=-\frac{32}{-4}$$

-32 ni -4 ga bo'lish.

$$x=8$$

Tenglama yechildi.

$$x=-\frac{3}{2} x=8$$

Kvadratni yakunlash bo'yicha qadamlar

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-2x^{2}+13x+24=0$$

Tenglamaning ikkala tarafidan 24 ni ayirish.

$$-2x^{2}+13x+24-24=-24$$

O‘zidan 24 ayirilsa 0 qoladi.

$$-2x^{2}+13x=-24$$

Ikki tarafini -2 ga bo‘ling.

$$\frac{-2x^{2}+13x}{-2}=-\frac{24}{-2}$$

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

$$x^{2}+\frac{13}{-2}x=-\frac{24}{-2}$$

13 ni -2 ga bo'lish.

$$x^{2}-\frac{13}{2}x=-\frac{24}{-2}$$

-24 ni -2 ga bo'lish.

$$x^{2}-\frac{13}{2}x=12$$

-\frac{13}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{13}{4} olish uchun. Keyin, -\frac{13}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{13}{2}x+\left(-\frac{13}{4}\right)^{2}=12+\left(-\frac{13}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{13}{4} kvadratini chiqarish.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=12+\frac{169}{16}$$

12 ni \frac{169}{16} ga qo'shish.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=\frac{361}{16}$$

x^{2}-\frac{13}{2}x+\frac{169}{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.

$$\left(x-\frac{13}{4}\right)^{2}=\frac{361}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{13}{4}\right)^{2}}=\sqrt{\frac{361}{16}}$$

Qisqartirish.

$$x-\frac{13}{4}=\frac{19}{4} x-\frac{13}{4}=-\frac{19}{4}$$

\frac{13}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=8 x=-\frac{3}{2}$$