2x^2-19x-10=0 eching

$$2 x ^ { 2 } - 19 x - 10 = 0$$

x uchun yechish

$$x=-\frac{1}{2}=-0,5$$

$$x=10$$

Guruhlash orqali faktorlash qadamlari

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2x^{2}+ax+bx-10 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-19 ab=2\left(-10\right)=-20$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -20-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-20 2,-10 4,-5$$

Har bir juftlik yigʻindisini hisoblang.

$$1-20=-19 2-10=-8 4-5=-1$$

Yechim – -19 yigʻindisini beruvchi juftlik.

$$a=-20 b=1$$

2x^{2}-19x-10 ni \left(2x^{2}-20x\right)+\left(x-10\right) sifatida qaytadan yozish.

$$\left(2x^{2}-20x\right)+\left(x-10\right)$$

2x^{2}-20x ichida 2x ni ajrating.

$$2x\left(x-10\right)+x-10$$

Distributiv funktsiyasidan foydalangan holda x-10 umumiy terminini chiqaring.

$$\left(x-10\right)\left(2x+1\right)$$

Tenglamani yechish uchun x-10=0 va 2x+1=0 ni yeching.

$$x=10 x=-\frac{1}{2}$$

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.

$$2x^{2}-19x-10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -19 ni b va -10 ni c bilan almashtiring.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 2\left(-10\right)}}{2\times 2}$$

-19 kvadratini chiqarish.

$$x=\frac{-\left(-19\right)±\sqrt{361-4\times 2\left(-10\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-19\right)±\sqrt{361-8\left(-10\right)}}{2\times 2}$$

-8 ni -10 marotabaga ko'paytirish.

$$x=\frac{-\left(-19\right)±\sqrt{361+80}}{2\times 2}$$

361 ni 80 ga qo'shish.

$$x=\frac{-\left(-19\right)±\sqrt{441}}{2\times 2}$$

441 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-19\right)±21}{2\times 2}$$

-19 ning teskarisi 19 ga teng.

$$x=\frac{19±21}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{19±21}{4}$$

x=\frac{19±21}{4} tenglamasini yeching, bunda ± musbat. 19 ni 21 ga qo'shish.

$$x=\frac{40}{4}$$

40 ni 4 ga bo'lish.

$$x=10$$

x=\frac{19±21}{4} tenglamasini yeching, bunda ± manfiy. 19 dan 21 ni ayirish.

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

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

$$x=-\frac{1}{2}$$

Tenglama yechildi.

$$x=10 x=-\frac{1}{2}$$

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.

$$2x^{2}-19x-10=0$$

10 ni tenglamaning ikkala tarafiga qo'shish.

$$2x^{2}-19x-10-\left(-10\right)=-\left(-10\right)$$

O‘zidan -10 ayirilsa 0 qoladi.

$$2x^{2}-19x=-\left(-10\right)$$

0 dan -10 ni ayirish.

$$2x^{2}-19x=10$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-19x}{2}=\frac{10}{2}$$

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

$$x^{2}-\frac{19}{2}x=\frac{10}{2}$$

10 ni 2 ga bo'lish.

$$x^{2}-\frac{19}{2}x=5$$

-\frac{19}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{19}{4} olish uchun. Keyin, -\frac{19}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{19}{2}x+\left(-\frac{19}{4}\right)^{2}=5+\left(-\frac{19}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{19}{4} kvadratini chiqarish.

$$x^{2}-\frac{19}{2}x+\frac{361}{16}=5+\frac{361}{16}$$

5 ni \frac{361}{16} ga qo'shish.

$$x^{2}-\frac{19}{2}x+\frac{361}{16}=\frac{441}{16}$$

x^{2}-\frac{19}{2}x+\frac{361}{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.

$$\left(x-\frac{19}{4}\right)^{2}=\frac{441}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{19}{4}\right)^{2}}=\sqrt{\frac{441}{16}}$$

Qisqartirish.

$$x-\frac{19}{4}=\frac{21}{4} x-\frac{19}{4}=-\frac{21}{4}$$

\frac{19}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=10 x=-\frac{1}{2}$$