16/x^2-9+4/x^2-5x+6-5(x+2)/6-x-x^2=0 eching

$$\frac { 16 } { x ^ { 2 } - 9 } + \frac { 4 } { x ^ { 2 } - 5 x + 6 } - \frac { 5 ( x + 2 ) } { 6 - x - x ^ { 2 } } = 0$$

x uchun yechish

$$x=-5$$

Guruhlash orqali faktorlash qadamlari

x qiymati -3,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x-2\right)\left(x+3\right) ga, x^{2}-9,x^{2}-5x+6,6-x-x^{2} ning eng kichik karralisiga ko‘paytiring.

$$\left(x-2\right)\times 16+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

x-2 ga 16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x-32+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

x+3 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x-32+4x+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

20x ni olish uchun 16x va 4x ni birlashtirish.

$$20x-32+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

-20 olish uchun -32 va 12'ni qo'shing.

$$20x-20-\left(3-x\right)\times 5\left(x+2\right)=0$$

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

$$20x-20-\left(15-5x\right)\left(x+2\right)=0$$

15-5x ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$20x-20-\left(5x+30-5x^{2}\right)=0$$

5x+30-5x^{2} teskarisini topish uchun har birining teskarisini toping.

$$20x-20-5x-30+5x^{2}=0$$

15x ni olish uchun 20x va -5x ni birlashtirish.

$$15x-20-30+5x^{2}=0$$

-50 olish uchun -20 dan 30 ni ayirish.

$$15x-50+5x^{2}=0$$

Ikki tarafini 5 ga bo‘ling.

$$3x-10+x^{2}=0$$

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

$$x^{2}+3x-10=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx-10 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=3 ab=1\left(-10\right)=-10$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -10-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,10 -2,5$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+10=9 -2+5=3$$

Yechim – 3 yigʻindisini beruvchi juftlik.

$$a=-2 b=5$$

x^{2}+3x-10 ni \left(x^{2}-2x\right)+\left(5x-10\right) sifatida qaytadan yozish.

$$\left(x^{2}-2x\right)+\left(5x-10\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$x\left(x-2\right)+5\left(x-2\right)$$

Distributiv funktsiyasidan foydalangan holda x-2 umumiy terminini chiqaring.

$$\left(x-2\right)\left(x+5\right)$$

Tenglamani yechish uchun x-2=0 va x+5=0 ni yeching.

$$x=2 x=-5$$

x qiymati 2 teng bo‘lmaydi.

$$x=-5$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -3,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x-2\right)\left(x+3\right) ga, x^{2}-9,x^{2}-5x+6,6-x-x^{2} ning eng kichik karralisiga ko‘paytiring.

$$\left(x-2\right)\times 16+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

x-2 ga 16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x-32+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

x+3 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x-32+4x+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

20x ni olish uchun 16x va 4x ni birlashtirish.

$$20x-32+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

-20 olish uchun -32 va 12'ni qo'shing.

$$20x-20-\left(3-x\right)\times 5\left(x+2\right)=0$$

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

$$20x-20-\left(15-5x\right)\left(x+2\right)=0$$

15-5x ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$20x-20-\left(5x+30-5x^{2}\right)=0$$

5x+30-5x^{2} teskarisini topish uchun har birining teskarisini toping.

$$20x-20-5x-30+5x^{2}=0$$

15x ni olish uchun 20x va -5x ni birlashtirish.

$$15x-20-30+5x^{2}=0$$

-50 olish uchun -20 dan 30 ni ayirish.

$$15x-50+5x^{2}=0$$

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.

$$5x^{2}+15x-50=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, 15 ni b va -50 ni c bilan almashtiring.

$$x=\frac{-15±\sqrt{15^{2}-4\times 5\left(-50\right)}}{2\times 5}$$

15 kvadratini chiqarish.

$$x=\frac{-15±\sqrt{225-4\times 5\left(-50\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-15±\sqrt{225-20\left(-50\right)}}{2\times 5}$$

-20 ni -50 marotabaga ko'paytirish.

$$x=\frac{-15±\sqrt{225+1000}}{2\times 5}$$

225 ni 1000 ga qo'shish.

$$x=\frac{-15±\sqrt{1225}}{2\times 5}$$

1225 ning kvadrat ildizini chiqarish.

$$x=\frac{-15±35}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{-15±35}{10}$$

x=\frac{-15±35}{10} tenglamasini yeching, bunda ± musbat. -15 ni 35 ga qo'shish.

$$x=\frac{20}{10}$$

20 ni 10 ga bo'lish.

$$x=2$$

x=\frac{-15±35}{10} tenglamasini yeching, bunda ± manfiy. -15 dan 35 ni ayirish.

$$x=-\frac{50}{10}$$

-50 ni 10 ga bo'lish.

$$x=-5$$

Tenglama yechildi.

$$x=2 x=-5$$

x qiymati 2 teng bo‘lmaydi.

$$x=-5$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -3,2,3 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-3\right)\left(x-2\right)\left(x+3\right) ga, x^{2}-9,x^{2}-5x+6,6-x-x^{2} ning eng kichik karralisiga ko‘paytiring.

$$\left(x-2\right)\times 16+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

x-2 ga 16 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x-32+\left(x+3\right)\times 4-\left(3-x\right)\times 5\left(x+2\right)=0$$

x+3 ga 4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x-32+4x+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

20x ni olish uchun 16x va 4x ni birlashtirish.

$$20x-32+12-\left(3-x\right)\times 5\left(x+2\right)=0$$

-20 olish uchun -32 va 12'ni qo'shing.

$$20x-20-\left(3-x\right)\times 5\left(x+2\right)=0$$

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

$$20x-20-\left(15-5x\right)\left(x+2\right)=0$$

15-5x ga x+2 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$20x-20-\left(5x+30-5x^{2}\right)=0$$

5x+30-5x^{2} teskarisini topish uchun har birining teskarisini toping.

$$20x-20-5x-30+5x^{2}=0$$

15x ni olish uchun 20x va -5x ni birlashtirish.

$$15x-20-30+5x^{2}=0$$

-50 olish uchun -20 dan 30 ni ayirish.

$$15x-50+5x^{2}=0$$

50 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$15x+5x^{2}=50$$

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

$$5x^{2}+15x=50$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}+15x}{5}=\frac{50}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{15}{5}x=\frac{50}{5}$$

15 ni 5 ga bo'lish.

$$x^{2}+3x=\frac{50}{5}$$

50 ni 5 ga bo'lish.

$$x^{2}+3x=10$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=10+\left(\frac{3}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{2} kvadratini chiqarish.

$$x^{2}+3x+\frac{9}{4}=10+\frac{9}{4}$$

10 ni \frac{9}{4} ga qo'shish.

$$x^{2}+3x+\frac{9}{4}=\frac{49}{4}$$

x^{2}+3x+\frac{9}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{49}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Qisqartirish.

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

Tenglamaning ikkala tarafidan \frac{3}{2} ni ayirish.

$$x=2 x=-5$$

x qiymati 2 teng bo‘lmaydi.

$$x=-5$$