(x+3)(6-x)/36-4x^2=-1/4 eching

$$\frac { ( x + 3 ) ( 6 - x ) } { 36 - 4 x ^ { 2 } } = - \frac { 1 } { 4 }$$

x uchun yechish

$$x = \frac{9}{2} = 4\frac{1}{2} = 4,5$$

Guruhlash orqali faktorlash qadamlari

x qiymati -3,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(x-3\right)\left(x+3\right) ga, 36-4x^{2},4 ning eng kichik karralisiga ko‘paytiring.

$$-\left(x+3\right)\left(6-x\right)=-\left(x-3\right)\left(x+3\right)$$

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

$$\left(-x-3\right)\left(6-x\right)=-\left(x-3\right)\left(x+3\right)$$

-x-3 ga 6-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-3x+x^{2}-18=-\left(x-3\right)\left(x+3\right)$$

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

$$-3x+x^{2}-18=\left(-x+3\right)\left(x+3\right)$$

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

$$-3x+x^{2}-18=-x^{2}+9$$

x^{2} ni ikki tarafga qo’shing.

$$-3x+x^{2}-18+x^{2}=9$$

2x^{2} ni olish uchun x^{2} va x^{2} ni birlashtirish.

$$-3x+2x^{2}-18=9$$

Ikkala tarafdan 9 ni ayirish.

$$-3x+2x^{2}-18-9=0$$

-27 olish uchun -18 dan 9 ni ayirish.

$$-3x+2x^{2}-27=0$$

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

$$2x^{2}-3x-27=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2x^{2}+ax+bx-27 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-3 ab=2\left(-27\right)=-54$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -54-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-54 2,-27 3,-18 6,-9$$

Har bir juftlik yigʻindisini hisoblang.

$$1-54=-53 2-27=-25 3-18=-15 6-9=-3$$

Yechim – -3 yigʻindisini beruvchi juftlik.

$$a=-9 b=6$$

2x^{2}-3x-27 ni \left(2x^{2}-9x\right)+\left(6x-27\right) sifatida qaytadan yozish.

$$\left(2x^{2}-9x\right)+\left(6x-27\right)$$

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

$$x\left(2x-9\right)+3\left(2x-9\right)$$

Distributiv funktsiyasidan foydalangan holda 2x-9 umumiy terminini chiqaring.

$$\left(2x-9\right)\left(x+3\right)$$

Tenglamani yechish uchun 2x-9=0 va x+3=0 ni yeching.

$$x=\frac{9}{2} x=-3$$

x qiymati -3 teng bo‘lmaydi.

$$x=\frac{9}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -3,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(x-3\right)\left(x+3\right) ga, 36-4x^{2},4 ning eng kichik karralisiga ko‘paytiring.

$$-\left(x+3\right)\left(6-x\right)=-\left(x-3\right)\left(x+3\right)$$

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

$$\left(-x-3\right)\left(6-x\right)=-\left(x-3\right)\left(x+3\right)$$

-x-3 ga 6-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-3x+x^{2}-18=-\left(x-3\right)\left(x+3\right)$$

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

$$-3x+x^{2}-18=\left(-x+3\right)\left(x+3\right)$$

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

$$-3x+x^{2}-18=-x^{2}+9$$

x^{2} ni ikki tarafga qo’shing.

$$-3x+x^{2}-18+x^{2}=9$$

2x^{2} ni olish uchun x^{2} va x^{2} ni birlashtirish.

$$-3x+2x^{2}-18=9$$

Ikkala tarafdan 9 ni ayirish.

$$-3x+2x^{2}-18-9=0$$

-27 olish uchun -18 dan 9 ni ayirish.

$$-3x+2x^{2}-27=0$$

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.

$$2x^{2}-3x-27=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -3 ni b va -27 ni c bilan almashtiring.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-27\right)}}{2\times 2}$$

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-27\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\left(-27\right)}}{2\times 2}$$

-8 ni -27 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9+216}}{2\times 2}$$

9 ni 216 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{225}}{2\times 2}$$

225 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-3\right)±15}{2\times 2}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±15}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{3±15}{4}$$

x=\frac{3±15}{4} tenglamasini yeching, bunda ± musbat. 3 ni 15 ga qo'shish.

$$x=\frac{18}{4}$$

\frac{18}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{9}{2}$$

x=\frac{3±15}{4} tenglamasini yeching, bunda ± manfiy. 3 dan 15 ni ayirish.

$$x=-\frac{12}{4}$$

-12 ni 4 ga bo'lish.

$$x=-3$$

Tenglama yechildi.

$$x=\frac{9}{2} x=-3$$

x qiymati -3 teng bo‘lmaydi.

$$x=\frac{9}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -3,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 4\left(x-3\right)\left(x+3\right) ga, 36-4x^{2},4 ning eng kichik karralisiga ko‘paytiring.

$$-\left(x+3\right)\left(6-x\right)=-\left(x-3\right)\left(x+3\right)$$

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

$$\left(-x-3\right)\left(6-x\right)=-\left(x-3\right)\left(x+3\right)$$

-x-3 ga 6-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-3x+x^{2}-18=-\left(x-3\right)\left(x+3\right)$$

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

$$-3x+x^{2}-18=\left(-x+3\right)\left(x+3\right)$$

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

$$-3x+x^{2}-18=-x^{2}+9$$

x^{2} ni ikki tarafga qo’shing.

$$-3x+x^{2}-18+x^{2}=9$$

2x^{2} ni olish uchun x^{2} va x^{2} ni birlashtirish.

$$-3x+2x^{2}-18=9$$

18 ni ikki tarafga qo’shing.

$$-3x+2x^{2}=9+18$$

27 olish uchun 9 va 18'ni qo'shing.

$$-3x+2x^{2}=27$$

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

$$2x^{2}-3x=27$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-3x}{2}=\frac{27}{2}$$

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

$$x^{2}-\frac{3}{2}x=\frac{27}{2}$$

-\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{4} olish uchun. Keyin, -\frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\frac{27}{2}+\left(-\frac{3}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{27}{2}+\frac{9}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{27}{2} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{225}{16}$$

x^{2}-\frac{3}{2}x+\frac{9}{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.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{225}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{225}{16}}$$

Qisqartirish.

$$x-\frac{3}{4}=\frac{15}{4} x-\frac{3}{4}=-\frac{15}{4}$$

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{9}{2} x=-3$$

x qiymati -3 teng bo‘lmaydi.

$$x=\frac{9}{2}$$