x+3x+1/2-x-2/3=x^2-2 eching

$$x + \frac { 3 x + 1 } { 2 } - \frac { x - 2 } { 3 } = x ^ { 2 } - 2$$

x uchun yechish

$$x=-1$$

$$x = \frac{19}{6} = 3\frac{1}{6} \approx 3,166666667$$

Guruhlash orqali faktorlash qadamlari

Tenglamaning ikkala tarafini 6 ga, 2,3 ning eng kichik karralisiga ko‘paytiring.

$$6x+3\left(3x+1\right)-2\left(x-2\right)=6x^{2}-12$$

3 ga 3x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6x+9x+3-2\left(x-2\right)=6x^{2}-12$$

15x ni olish uchun 6x va 9x ni birlashtirish.

$$15x+3-2\left(x-2\right)=6x^{2}-12$$

-2 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$15x+3-2x+4=6x^{2}-12$$

13x ni olish uchun 15x va -2x ni birlashtirish.

$$13x+3+4=6x^{2}-12$$

7 olish uchun 3 va 4'ni qo'shing.

$$13x+7=6x^{2}-12$$

Ikkala tarafdan 6x^{2} ni ayirish.

$$13x+7-6x^{2}=-12$$

12 ni ikki tarafga qo’shing.

$$13x+7-6x^{2}+12=0$$

19 olish uchun 7 va 12'ni qo'shing.

$$13x+19-6x^{2}=0$$

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

$$-6x^{2}+13x+19=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -6x^{2}+ax+bx+19 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=13 ab=-6\times 19=-114$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -114-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,114 -2,57 -3,38 -6,19$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+114=113 -2+57=55 -3+38=35 -6+19=13$$

Yechim – 13 yigʻindisini beruvchi juftlik.

$$a=19 b=-6$$

-6x^{2}+13x+19 ni \left(-6x^{2}+19x\right)+\left(-6x+19\right) sifatida qaytadan yozish.

$$\left(-6x^{2}+19x\right)+\left(-6x+19\right)$$

Birinchi guruhda -x ni va ikkinchi guruhda -1 ni faktordan chiqaring.

$$-x\left(6x-19\right)-\left(6x-19\right)$$

Distributiv funktsiyasidan foydalangan holda 6x-19 umumiy terminini chiqaring.

$$\left(6x-19\right)\left(-x-1\right)$$

Tenglamani yechish uchun 6x-19=0 va -x-1=0 ni yeching.

$$x=\frac{19}{6} x=-1$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 6 ga, 2,3 ning eng kichik karralisiga ko‘paytiring.

$$6x+3\left(3x+1\right)-2\left(x-2\right)=6x^{2}-12$$

3 ga 3x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6x+9x+3-2\left(x-2\right)=6x^{2}-12$$

15x ni olish uchun 6x va 9x ni birlashtirish.

$$15x+3-2\left(x-2\right)=6x^{2}-12$$

-2 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$15x+3-2x+4=6x^{2}-12$$

13x ni olish uchun 15x va -2x ni birlashtirish.

$$13x+3+4=6x^{2}-12$$

7 olish uchun 3 va 4'ni qo'shing.

$$13x+7=6x^{2}-12$$

Ikkala tarafdan 6x^{2} ni ayirish.

$$13x+7-6x^{2}=-12$$

12 ni ikki tarafga qo’shing.

$$13x+7-6x^{2}+12=0$$

19 olish uchun 7 va 12'ni qo'shing.

$$13x+19-6x^{2}=0$$

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.

$$-6x^{2}+13x+19=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -6 ni a, 13 ni b va 19 ni c bilan almashtiring.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-6\right)\times 19}}{2\left(-6\right)}$$

13 kvadratini chiqarish.

$$x=\frac{-13±\sqrt{169-4\left(-6\right)\times 19}}{2\left(-6\right)}$$

-4 ni -6 marotabaga ko'paytirish.

$$x=\frac{-13±\sqrt{169+24\times 19}}{2\left(-6\right)}$$

24 ni 19 marotabaga ko'paytirish.

$$x=\frac{-13±\sqrt{169+456}}{2\left(-6\right)}$$

169 ni 456 ga qo'shish.

$$x=\frac{-13±\sqrt{625}}{2\left(-6\right)}$$

625 ning kvadrat ildizini chiqarish.

$$x=\frac{-13±25}{2\left(-6\right)}$$

2 ni -6 marotabaga ko'paytirish.

$$x=\frac{-13±25}{-12}$$

x=\frac{-13±25}{-12} tenglamasini yeching, bunda ± musbat. -13 ni 25 ga qo'shish.

$$x=\frac{12}{-12}$$

12 ni -12 ga bo'lish.

$$x=-1$$

x=\frac{-13±25}{-12} tenglamasini yeching, bunda ± manfiy. -13 dan 25 ni ayirish.

$$x=-\frac{38}{-12}$$

\frac{-38}{-12} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{19}{6}$$

Tenglama yechildi.

$$x=-1 x=\frac{19}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 6 ga, 2,3 ning eng kichik karralisiga ko‘paytiring.

$$6x+3\left(3x+1\right)-2\left(x-2\right)=6x^{2}-12$$

3 ga 3x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6x+9x+3-2\left(x-2\right)=6x^{2}-12$$

15x ni olish uchun 6x va 9x ni birlashtirish.

$$15x+3-2\left(x-2\right)=6x^{2}-12$$

-2 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$15x+3-2x+4=6x^{2}-12$$

13x ni olish uchun 15x va -2x ni birlashtirish.

$$13x+3+4=6x^{2}-12$$

7 olish uchun 3 va 4'ni qo'shing.

$$13x+7=6x^{2}-12$$

Ikkala tarafdan 6x^{2} ni ayirish.

$$13x+7-6x^{2}=-12$$

Ikkala tarafdan 7 ni ayirish.

$$13x-6x^{2}=-12-7$$

-19 olish uchun -12 dan 7 ni ayirish.

$$13x-6x^{2}=-19$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-6x^{2}+13x=-19$$

Ikki tarafini -6 ga bo‘ling.

$$\frac{-6x^{2}+13x}{-6}=-\frac{19}{-6}$$

-6 ga bo'lish -6 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{13}{-6}x=-\frac{19}{-6}$$

13 ni -6 ga bo'lish.

$$x^{2}-\frac{13}{6}x=-\frac{19}{-6}$$

-19 ni -6 ga bo'lish.

$$x^{2}-\frac{13}{6}x=\frac{19}{6}$$

-\frac{13}{6} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{13}{12} olish uchun. Keyin, -\frac{13}{12} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{13}{6}x+\left(-\frac{13}{12}\right)^{2}=\frac{19}{6}+\left(-\frac{13}{12}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{13}{12} kvadratini chiqarish.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{19}{6}+\frac{169}{144}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{19}{6} ni \frac{169}{144} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{13}{6}x+\frac{169}{144}=\frac{625}{144}$$

x^{2}-\frac{13}{6}x+\frac{169}{144} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{13}{12}\right)^{2}=\frac{625}{144}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{13}{12}\right)^{2}}=\sqrt{\frac{625}{144}}$$

Qisqartirish.

$$x-\frac{13}{12}=\frac{25}{12} x-\frac{13}{12}=-\frac{25}{12}$$

\frac{13}{12} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{19}{6} x=-1$$