2x^2-5x-18=0 eching

$$2 { x }^{ 2 } -5x-18 = 0$$

x uchun yechish

$$x=-2$$

$$x = \frac{9}{2} = 4\frac{1}{2} = 4,5$$

Guruhlash orqali faktorlash qadamlari

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2x^{2}+ax+bx-18 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-5 ab=2\left(-18\right)=-36$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -36-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Har bir juftlik yigʻindisini hisoblang.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

Yechim – -5 yigʻindisini beruvchi juftlik.

$$a=-9 b=4$$

2x^{2}-5x-18 ni \left(2x^{2}-9x\right)+\left(4x-18\right) sifatida qaytadan yozish.

$$\left(2x^{2}-9x\right)+\left(4x-18\right)$$

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

$$x\left(2x-9\right)+2\left(2x-9\right)$$

Distributiv funktsiyasidan foydalangan holda 2x-9 umumiy terminini chiqaring.

$$\left(2x-9\right)\left(x+2\right)$$

Tenglamani yechish uchun 2x-9=0 va x+2=0 ni yeching.

$$x=\frac{9}{2} x=-2$$

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.

$$2x^{2}-5x-18=0$$

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 -18 ni c bilan almashtiring.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 2\left(-18\right)}}{2\times 2}$$

-5 kvadratini chiqarish.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 2\left(-18\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25-8\left(-18\right)}}{2\times 2}$$

-8 ni -18 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25+144}}{2\times 2}$$

25 ni 144 ga qo'shish.

$$x=\frac{-\left(-5\right)±\sqrt{169}}{2\times 2}$$

169 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-5\right)±13}{2\times 2}$$

-5 ning teskarisi 5 ga teng.

$$x=\frac{5±13}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{5±13}{4}$$

x=\frac{5±13}{4} tenglamasini yeching, bunda ± musbat. 5 ni 13 ga qo'shish.

$$x=\frac{18}{4}$$

\frac{18}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{9}{2}$$

x=\frac{5±13}{4} tenglamasini yeching, bunda ± manfiy. 5 dan 13 ni ayirish.

$$x=-\frac{8}{4}$$

-8 ni 4 ga bo'lish.

$$x=-2$$

Tenglama yechildi.

$$x=\frac{9}{2} x=-2$$

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.

$$2x^{2}-5x-18=0$$

18 ni tenglamaning ikkala tarafiga qo'shish.

$$2x^{2}-5x-18-\left(-18\right)=-\left(-18\right)$$

O‘zidan -18 ayirilsa 0 qoladi.

$$2x^{2}-5x=-\left(-18\right)$$

0 dan -18 ni ayirish.

$$2x^{2}-5x=18$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-5x}{2}=\frac{18}{2}$$

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

$$x^{2}-\frac{5}{2}x=\frac{18}{2}$$

18 ni 2 ga bo'lish.

$$x^{2}-\frac{5}{2}x=9$$

-\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.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=9+\left(-\frac{5}{4}\right)^{2}$$

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

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=9+\frac{25}{16}$$

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

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{169}{16}$$

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.

$$\left(x-\frac{5}{4}\right)^{2}=\frac{169}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{\frac{169}{16}}$$

Qisqartirish.

$$x-\frac{5}{4}=\frac{13}{4} x-\frac{5}{4}=-\frac{13}{4}$$

\frac{5}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{9}{2} x=-2$$