x^2-9=(2x+6)(7x-2) eching

$$x ^ { 2 } - 9 = ( 2 x + 6 ) ( 7 x - 2 )$$

x uchun yechish

$$x=\frac{1}{13}\approx 0,076923077$$

$$x=-3$$

Guruhlash orqali faktorlash qadamlari

2x+6 ga 7x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}-9=14x^{2}+38x-12$$

Ikkala tarafdan 14x^{2} ni ayirish.

$$x^{2}-9-14x^{2}=38x-12$$

-13x^{2} ni olish uchun x^{2} va -14x^{2} ni birlashtirish.

$$-13x^{2}-9=38x-12$$

Ikkala tarafdan 38x ni ayirish.

$$-13x^{2}-9-38x=-12$$

12 ni ikki tarafga qo’shing.

$$-13x^{2}-9-38x+12=0$$

3 olish uchun -9 va 12'ni qo'shing.

$$-13x^{2}+3-38x=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$-13x^{2}-38x+3=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -13x^{2}+ax+bx+3 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-38 ab=-13\times 3=-39$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -39-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-39 3,-13$$

Har bir juftlik yigʻindisini hisoblang.

$$1-39=-38 3-13=-10$$

Yechim – -38 yigʻindisini beruvchi juftlik.

$$a=1 b=-39$$

-13x^{2}-38x+3 ni \left(-13x^{2}+x\right)+\left(-39x+3\right) sifatida qaytadan yozish.

$$\left(-13x^{2}+x\right)+\left(-39x+3\right)$$

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

$$-x\left(13x-1\right)-3\left(13x-1\right)$$

Distributiv funktsiyasidan foydalangan holda 13x-1 umumiy terminini chiqaring.

$$\left(13x-1\right)\left(-x-3\right)$$

Tenglamani yechish uchun 13x-1=0 va -x-3=0 ni yeching.

$$x=\frac{1}{13} x=-3$$

Kvadrat tenglamalaridan foydalanish qoidalari

2x+6 ga 7x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}-9=14x^{2}+38x-12$$

Ikkala tarafdan 14x^{2} ni ayirish.

$$x^{2}-9-14x^{2}=38x-12$$

-13x^{2} ni olish uchun x^{2} va -14x^{2} ni birlashtirish.

$$-13x^{2}-9=38x-12$$

Ikkala tarafdan 38x ni ayirish.

$$-13x^{2}-9-38x=-12$$

12 ni ikki tarafga qo’shing.

$$-13x^{2}-9-38x+12=0$$

3 olish uchun -9 va 12'ni qo'shing.

$$-13x^{2}+3-38x=0$$

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.

$$-13x^{2}-38x+3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -13 ni a, -38 ni b va 3 ni c bilan almashtiring.

$$x=\frac{-\left(-38\right)±\sqrt{\left(-38\right)^{2}-4\left(-13\right)\times 3}}{2\left(-13\right)}$$

-38 kvadratini chiqarish.

$$x=\frac{-\left(-38\right)±\sqrt{1444-4\left(-13\right)\times 3}}{2\left(-13\right)}$$

-4 ni -13 marotabaga ko'paytirish.

$$x=\frac{-\left(-38\right)±\sqrt{1444+52\times 3}}{2\left(-13\right)}$$

52 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-38\right)±\sqrt{1444+156}}{2\left(-13\right)}$$

1444 ni 156 ga qo'shish.

$$x=\frac{-\left(-38\right)±\sqrt{1600}}{2\left(-13\right)}$$

1600 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-38\right)±40}{2\left(-13\right)}$$

-38 ning teskarisi 38 ga teng.

$$x=\frac{38±40}{2\left(-13\right)}$$

2 ni -13 marotabaga ko'paytirish.

$$x=\frac{38±40}{-26}$$

x=\frac{38±40}{-26} tenglamasini yeching, bunda ± musbat. 38 ni 40 ga qo'shish.

$$x=\frac{78}{-26}$$

78 ni -26 ga bo'lish.

$$x=-3$$

x=\frac{38±40}{-26} tenglamasini yeching, bunda ± manfiy. 38 dan 40 ni ayirish.

$$x=-\frac{2}{-26}$$

\frac{-2}{-26} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{1}{13}$$

Tenglama yechildi.

$$x=-3 x=\frac{1}{13}$$

Kvadratni yakunlash bo'yicha qadamlar

2x+6 ga 7x-2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}-9=14x^{2}+38x-12$$

Ikkala tarafdan 14x^{2} ni ayirish.

$$x^{2}-9-14x^{2}=38x-12$$

-13x^{2} ni olish uchun x^{2} va -14x^{2} ni birlashtirish.

$$-13x^{2}-9=38x-12$$

Ikkala tarafdan 38x ni ayirish.

$$-13x^{2}-9-38x=-12$$

9 ni ikki tarafga qo’shing.

$$-13x^{2}-38x=-12+9$$

-3 olish uchun -12 va 9'ni qo'shing.

$$-13x^{2}-38x=-3$$

Ikki tarafini -13 ga bo‘ling.

$$\frac{-13x^{2}-38x}{-13}=-\frac{3}{-13}$$

-13 ga bo'lish -13 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{38}{-13}\right)x=-\frac{3}{-13}$$

-38 ni -13 ga bo'lish.

$$x^{2}+\frac{38}{13}x=-\frac{3}{-13}$$

-3 ni -13 ga bo'lish.

$$x^{2}+\frac{38}{13}x=\frac{3}{13}$$

\frac{38}{13} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{19}{13} olish uchun. Keyin, \frac{19}{13} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{38}{13}x+\left(\frac{19}{13}\right)^{2}=\frac{3}{13}+\left(\frac{19}{13}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{19}{13} kvadratini chiqarish.

$$x^{2}+\frac{38}{13}x+\frac{361}{169}=\frac{3}{13}+\frac{361}{169}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{13} ni \frac{361}{169} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{38}{13}x+\frac{361}{169}=\frac{400}{169}$$

x^{2}+\frac{38}{13}x+\frac{361}{169} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{19}{13}\right)^{2}=\frac{400}{169}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{19}{13}\right)^{2}}=\sqrt{\frac{400}{169}}$$

Qisqartirish.

$$x+\frac{19}{13}=\frac{20}{13} x+\frac{19}{13}=-\frac{20}{13}$$

Tenglamaning ikkala tarafidan \frac{19}{13} ni ayirish.

$$x=\frac{1}{13} x=-3$$