2x^2+5x-168=0 eching

$$2 { x }^{ 2 } +5x-168=0$$

x uchun yechish

$$x = -\frac{21}{2} = -10\frac{1}{2} = -10,5$$

$$x=8$$

Guruhlash orqali faktorlash qadamlari

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2x^{2}+ax+bx-168 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=5 ab=2\left(-168\right)=-336$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -336-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,336 -2,168 -3,112 -4,84 -6,56 -7,48 -8,42 -12,28 -14,24 -16,21$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+336=335 -2+168=166 -3+112=109 -4+84=80 -6+56=50 -7+48=41 -8+42=34 -12+28=16 -14+24=10 -16+21=5$$

Yechim – 5 yigʻindisini beruvchi juftlik.

$$a=-16 b=21$$

2x^{2}+5x-168 ni \left(2x^{2}-16x\right)+\left(21x-168\right) sifatida qaytadan yozish.

$$\left(2x^{2}-16x\right)+\left(21x-168\right)$$

Birinchi guruhda 2x ni va ikkinchi guruhda 21 ni faktordan chiqaring.

$$2x\left(x-8\right)+21\left(x-8\right)$$

Distributiv funktsiyasidan foydalangan holda x-8 umumiy terminini chiqaring.

$$\left(x-8\right)\left(2x+21\right)$$

Tenglamani yechish uchun x-8=0 va 2x+21=0 ni yeching.

$$x=8 x=-\frac{21}{2}$$

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.

$$2x^{2}+5x-168=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 5 ni b va -168 ni c bilan almashtiring.

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\left(-168\right)}}{2\times 2}$$

5 kvadratini chiqarish.

$$x=\frac{-5±\sqrt{25-4\times 2\left(-168\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25-8\left(-168\right)}}{2\times 2}$$

-8 ni -168 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25+1344}}{2\times 2}$$

25 ni 1344 ga qo'shish.

$$x=\frac{-5±\sqrt{1369}}{2\times 2}$$

1369 ning kvadrat ildizini chiqarish.

$$x=\frac{-5±37}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-5±37}{4}$$

x=\frac{-5±37}{4} tenglamasini yeching, bunda ± musbat. -5 ni 37 ga qo'shish.

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

32 ni 4 ga bo'lish.

$$x=8$$

x=\frac{-5±37}{4} tenglamasini yeching, bunda ± manfiy. -5 dan 37 ni ayirish.

$$x=-\frac{42}{4}$$

\frac{-42}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{21}{2}$$

Tenglama yechildi.

$$x=8 x=-\frac{21}{2}$$

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.

$$2x^{2}+5x-168=0$$

168 ni tenglamaning ikkala tarafiga qo'shish.

$$2x^{2}+5x-168-\left(-168\right)=-\left(-168\right)$$

O‘zidan -168 ayirilsa 0 qoladi.

$$2x^{2}+5x=-\left(-168\right)$$

0 dan -168 ni ayirish.

$$2x^{2}+5x=168$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+5x}{2}=\frac{168}{2}$$

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

$$x^{2}+\frac{5}{2}x=\frac{168}{2}$$

168 ni 2 ga bo'lish.

$$x^{2}+\frac{5}{2}x=84$$

\frac{5}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{4} olish uchun. Keyin, \frac{5}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=84+\left(\frac{5}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{4} kvadratini chiqarish.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=84+\frac{25}{16}$$

84 ni \frac{25}{16} ga qo'shish.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{1369}{16}$$

x^{2}+\frac{5}{2}x+\frac{25}{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.

$$\left(x+\frac{5}{4}\right)^{2}=\frac{1369}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{\frac{1369}{16}}$$

Qisqartirish.

$$x+\frac{5}{4}=\frac{37}{4} x+\frac{5}{4}=-\frac{37}{4}$$

Tenglamaning ikkala tarafidan \frac{5}{4} ni ayirish.

$$x=8 x=-\frac{21}{2}$$