x^2+6x-7/3x^2-x-2=5 eching

$$\frac { x ^ { 2 } + 6 x - 7 } { 3 x ^ { 2 } - x - 2 } = 5$$

x uchun yechish

$$x=-\frac{3}{14}\approx -0,214285714$$

Guruhlash orqali faktorlash qadamlari

x qiymati -\frac{2}{3},1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(3x+2\right) ga ko'paytirish.

$$x^{2}+6x-7=5\left(x-1\right)\left(3x+2\right)$$

5 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}+6x-7=\left(5x-5\right)\left(3x+2\right)$$

5x-5 ga 3x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}+6x-7=15x^{2}-5x-10$$

Ikkala tarafdan 15x^{2} ni ayirish.

$$x^{2}+6x-7-15x^{2}=-5x-10$$

-14x^{2} ni olish uchun x^{2} va -15x^{2} ni birlashtirish.

$$-14x^{2}+6x-7=-5x-10$$

5x ni ikki tarafga qo’shing.

$$-14x^{2}+6x-7+5x=-10$$

11x ni olish uchun 6x va 5x ni birlashtirish.

$$-14x^{2}+11x-7=-10$$

10 ni ikki tarafga qo’shing.

$$-14x^{2}+11x-7+10=0$$

3 olish uchun -7 va 10'ni qo'shing.

$$-14x^{2}+11x+3=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -14x^{2}+ax+bx+3 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=11 ab=-14\times 3=-42$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -42-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,42 -2,21 -3,14 -6,7$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+42=41 -2+21=19 -3+14=11 -6+7=1$$

Yechim – 11 yigʻindisini beruvchi juftlik.

$$a=14 b=-3$$

-14x^{2}+11x+3 ni \left(-14x^{2}+14x\right)+\left(-3x+3\right) sifatida qaytadan yozish.

$$\left(-14x^{2}+14x\right)+\left(-3x+3\right)$$

Birinchi guruhda 14x ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$14x\left(-x+1\right)+3\left(-x+1\right)$$

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

$$\left(-x+1\right)\left(14x+3\right)$$

Tenglamani yechish uchun -x+1=0 va 14x+3=0 ni yeching.

$$x=1 x=-\frac{3}{14}$$

x qiymati 1 teng bo‘lmaydi.

$$x=-\frac{3}{14}$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -\frac{2}{3},1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(3x+2\right) ga ko'paytirish.

$$x^{2}+6x-7=5\left(x-1\right)\left(3x+2\right)$$

5 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}+6x-7=\left(5x-5\right)\left(3x+2\right)$$

5x-5 ga 3x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}+6x-7=15x^{2}-5x-10$$

Ikkala tarafdan 15x^{2} ni ayirish.

$$x^{2}+6x-7-15x^{2}=-5x-10$$

-14x^{2} ni olish uchun x^{2} va -15x^{2} ni birlashtirish.

$$-14x^{2}+6x-7=-5x-10$$

5x ni ikki tarafga qo’shing.

$$-14x^{2}+6x-7+5x=-10$$

11x ni olish uchun 6x va 5x ni birlashtirish.

$$-14x^{2}+11x-7=-10$$

10 ni ikki tarafga qo’shing.

$$-14x^{2}+11x-7+10=0$$

3 olish uchun -7 va 10'ni qo'shing.

$$-14x^{2}+11x+3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -14 ni a, 11 ni b va 3 ni c bilan almashtiring.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-14\right)\times 3}}{2\left(-14\right)}$$

11 kvadratini chiqarish.

$$x=\frac{-11±\sqrt{121-4\left(-14\right)\times 3}}{2\left(-14\right)}$$

-4 ni -14 marotabaga ko'paytirish.

$$x=\frac{-11±\sqrt{121+56\times 3}}{2\left(-14\right)}$$

56 ni 3 marotabaga ko'paytirish.

$$x=\frac{-11±\sqrt{121+168}}{2\left(-14\right)}$$

121 ni 168 ga qo'shish.

$$x=\frac{-11±\sqrt{289}}{2\left(-14\right)}$$

289 ning kvadrat ildizini chiqarish.

$$x=\frac{-11±17}{2\left(-14\right)}$$

2 ni -14 marotabaga ko'paytirish.

$$x=\frac{-11±17}{-28}$$

x=\frac{-11±17}{-28} tenglamasini yeching, bunda ± musbat. -11 ni 17 ga qo'shish.

$$x=\frac{6}{-28}$$

\frac{6}{-28} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{3}{14}$$

x=\frac{-11±17}{-28} tenglamasini yeching, bunda ± manfiy. -11 dan 17 ni ayirish.

$$x=-\frac{28}{-28}$$

-28 ni -28 ga bo'lish.

$$x=1$$

Tenglama yechildi.

$$x=-\frac{3}{14} x=1$$

x qiymati 1 teng bo‘lmaydi.

$$x=-\frac{3}{14}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -\frac{2}{3},1 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-1\right)\left(3x+2\right) ga ko'paytirish.

$$x^{2}+6x-7=5\left(x-1\right)\left(3x+2\right)$$

5 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}+6x-7=\left(5x-5\right)\left(3x+2\right)$$

5x-5 ga 3x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}+6x-7=15x^{2}-5x-10$$

Ikkala tarafdan 15x^{2} ni ayirish.

$$x^{2}+6x-7-15x^{2}=-5x-10$$

-14x^{2} ni olish uchun x^{2} va -15x^{2} ni birlashtirish.

$$-14x^{2}+6x-7=-5x-10$$

5x ni ikki tarafga qo’shing.

$$-14x^{2}+6x-7+5x=-10$$

11x ni olish uchun 6x va 5x ni birlashtirish.

$$-14x^{2}+11x-7=-10$$

7 ni ikki tarafga qo’shing.

$$-14x^{2}+11x=-10+7$$

-3 olish uchun -10 va 7'ni qo'shing.

$$-14x^{2}+11x=-3$$

Ikki tarafini -14 ga bo‘ling.

$$\frac{-14x^{2}+11x}{-14}=-\frac{3}{-14}$$

-14 ga bo'lish -14 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{11}{-14}x=-\frac{3}{-14}$$

11 ni -14 ga bo'lish.

$$x^{2}-\frac{11}{14}x=-\frac{3}{-14}$$

-3 ni -14 ga bo'lish.

$$x^{2}-\frac{11}{14}x=\frac{3}{14}$$

-\frac{11}{14} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{28} olish uchun. Keyin, -\frac{11}{28} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{11}{14}x+\left(-\frac{11}{28}\right)^{2}=\frac{3}{14}+\left(-\frac{11}{28}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{28} kvadratini chiqarish.

$$x^{2}-\frac{11}{14}x+\frac{121}{784}=\frac{3}{14}+\frac{121}{784}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{14} ni \frac{121}{784} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{11}{14}x+\frac{121}{784}=\frac{289}{784}$$

x^{2}-\frac{11}{14}x+\frac{121}{784} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{11}{28}\right)^{2}=\frac{289}{784}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{28}\right)^{2}}=\sqrt{\frac{289}{784}}$$

Qisqartirish.

$$x-\frac{11}{28}=\frac{17}{28} x-\frac{11}{28}=-\frac{17}{28}$$

\frac{11}{28} ni tenglamaning ikkala tarafiga qo'shish.

$$x=1 x=-\frac{3}{14}$$

x qiymati 1 teng bo‘lmaydi.

$$x=-\frac{3}{14}$$