19x^2+15x-34=0 eching

$$19 { x }^{ 2 } +15x-34=0$$

x uchun yechish

$$x = -\frac{34}{19} = -1\frac{15}{19} \approx -1,789473684$$

$$x=1$$

Guruhlash orqali faktorlash qadamlari

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 19x^{2}+ax+bx-34 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=15 ab=19\left(-34\right)=-646$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -646-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,646 -2,323 -17,38 -19,34$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+646=645 -2+323=321 -17+38=21 -19+34=15$$

Yechim – 15 yigʻindisini beruvchi juftlik.

$$a=-19 b=34$$

19x^{2}+15x-34 ni \left(19x^{2}-19x\right)+\left(34x-34\right) sifatida qaytadan yozish.

$$\left(19x^{2}-19x\right)+\left(34x-34\right)$$

Birinchi guruhda 19x ni va ikkinchi guruhda 34 ni faktordan chiqaring.

$$19x\left(x-1\right)+34\left(x-1\right)$$

Distributiv funktsiyasidan foydalangan holda x-1 umumiy terminini chiqaring.

$$\left(x-1\right)\left(19x+34\right)$$

Tenglamani yechish uchun x-1=0 va 19x+34=0 ni yeching.

$$x=1 x=-\frac{34}{19}$$

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.

$$19x^{2}+15x-34=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 19 ni a, 15 ni b va -34 ni c bilan almashtiring.

$$x=\frac{-15±\sqrt{15^{2}-4\times 19\left(-34\right)}}{2\times 19}$$

15 kvadratini chiqarish.

$$x=\frac{-15±\sqrt{225-4\times 19\left(-34\right)}}{2\times 19}$$

-4 ni 19 marotabaga ko'paytirish.

$$x=\frac{-15±\sqrt{225-76\left(-34\right)}}{2\times 19}$$

-76 ni -34 marotabaga ko'paytirish.

$$x=\frac{-15±\sqrt{225+2584}}{2\times 19}$$

225 ni 2584 ga qo'shish.

$$x=\frac{-15±\sqrt{2809}}{2\times 19}$$

2809 ning kvadrat ildizini chiqarish.

$$x=\frac{-15±53}{2\times 19}$$

2 ni 19 marotabaga ko'paytirish.

$$x=\frac{-15±53}{38}$$

x=\frac{-15±53}{38} tenglamasini yeching, bunda ± musbat. -15 ni 53 ga qo'shish.

$$x=\frac{38}{38}$$

38 ni 38 ga bo'lish.

$$x=1$$

x=\frac{-15±53}{38} tenglamasini yeching, bunda ± manfiy. -15 dan 53 ni ayirish.

$$x=-\frac{68}{38}$$

\frac{-68}{38} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{34}{19}$$

Tenglama yechildi.

$$x=1 x=-\frac{34}{19}$$

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.

$$19x^{2}+15x-34=0$$

34 ni tenglamaning ikkala tarafiga qo'shish.

$$19x^{2}+15x-34-\left(-34\right)=-\left(-34\right)$$

O‘zidan -34 ayirilsa 0 qoladi.

$$19x^{2}+15x=-\left(-34\right)$$

0 dan -34 ni ayirish.

$$19x^{2}+15x=34$$

Ikki tarafini 19 ga bo‘ling.

$$\frac{19x^{2}+15x}{19}=\frac{34}{19}$$

19 ga bo'lish 19 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{15}{19}x=\frac{34}{19}$$

\frac{15}{19} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{15}{38} olish uchun. Keyin, \frac{15}{38} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{15}{19}x+\left(\frac{15}{38}\right)^{2}=\frac{34}{19}+\left(\frac{15}{38}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{15}{38} kvadratini chiqarish.

$$x^{2}+\frac{15}{19}x+\frac{225}{1444}=\frac{34}{19}+\frac{225}{1444}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{34}{19} ni \frac{225}{1444} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{15}{19}x+\frac{225}{1444}=\frac{2809}{1444}$$

x^{2}+\frac{15}{19}x+\frac{225}{1444} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{15}{38}\right)^{2}=\frac{2809}{1444}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{15}{38}\right)^{2}}=\sqrt{\frac{2809}{1444}}$$

Qisqartirish.

$$x+\frac{15}{38}=\frac{53}{38} x+\frac{15}{38}=-\frac{53}{38}$$

Tenglamaning ikkala tarafidan \frac{15}{38} ni ayirish.

$$x=1 x=-\frac{34}{19}$$