12(n-4)-30=n^2-9n+12 eching

$$12 ( n - 4 ) - 30 = n ^ { 2 } - 9 n + 12$$

n uchun yechish

$$n=6$$

$$n=15$$

Guruhlash orqali faktorlash qadamlari

12 ga n-4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$12n-48-30=n^{2}-9n+12$$

-78 olish uchun -48 dan 30 ni ayirish.

$$12n-78=n^{2}-9n+12$$

Ikkala tarafdan n^{2} ni ayirish.

$$12n-78-n^{2}=-9n+12$$

9n ni ikki tarafga qo’shing.

$$12n-78-n^{2}+9n=12$$

21n ni olish uchun 12n va 9n ni birlashtirish.

$$21n-78-n^{2}=12$$

Ikkala tarafdan 12 ni ayirish.

$$21n-78-n^{2}-12=0$$

-90 olish uchun -78 dan 12 ni ayirish.

$$21n-90-n^{2}=0$$

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

$$-n^{2}+21n-90=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -n^{2}+an+bn-90 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=21 ab=-\left(-90\right)=90$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 90-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,90 2,45 3,30 5,18 6,15 9,10$$

Har bir juftlik yigʻindisini hisoblang.

$$1+90=91 2+45=47 3+30=33 5+18=23 6+15=21 9+10=19$$

Yechim – 21 yigʻindisini beruvchi juftlik.

$$a=15 b=6$$

-n^{2}+21n-90 ni \left(-n^{2}+15n\right)+\left(6n-90\right) sifatida qaytadan yozish.

$$\left(-n^{2}+15n\right)+\left(6n-90\right)$$

Birinchi guruhda -n ni va ikkinchi guruhda 6 ni faktordan chiqaring.

$$-n\left(n-15\right)+6\left(n-15\right)$$

Distributiv funktsiyasidan foydalangan holda n-15 umumiy terminini chiqaring.

$$\left(n-15\right)\left(-n+6\right)$$

Tenglamani yechish uchun n-15=0 va -n+6=0 ni yeching.

$$n=15 n=6$$

Kvadrat tenglamalaridan foydalanish qoidalari

12 ga n-4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$12n-48-30=n^{2}-9n+12$$

-78 olish uchun -48 dan 30 ni ayirish.

$$12n-78=n^{2}-9n+12$$

Ikkala tarafdan n^{2} ni ayirish.

$$12n-78-n^{2}=-9n+12$$

9n ni ikki tarafga qo’shing.

$$12n-78-n^{2}+9n=12$$

21n ni olish uchun 12n va 9n ni birlashtirish.

$$21n-78-n^{2}=12$$

Ikkala tarafdan 12 ni ayirish.

$$21n-78-n^{2}-12=0$$

-90 olish uchun -78 dan 12 ni ayirish.

$$21n-90-n^{2}=0$$

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.

$$-n^{2}+21n-90=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 21 ni b va -90 ni c bilan almashtiring.

$$n=\frac{-21±\sqrt{21^{2}-4\left(-1\right)\left(-90\right)}}{2\left(-1\right)}$$

21 kvadratini chiqarish.

$$n=\frac{-21±\sqrt{441-4\left(-1\right)\left(-90\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$n=\frac{-21±\sqrt{441+4\left(-90\right)}}{2\left(-1\right)}$$

4 ni -90 marotabaga ko'paytirish.

$$n=\frac{-21±\sqrt{441-360}}{2\left(-1\right)}$$

441 ni -360 ga qo'shish.

$$n=\frac{-21±\sqrt{81}}{2\left(-1\right)}$$

81 ning kvadrat ildizini chiqarish.

$$n=\frac{-21±9}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$n=\frac{-21±9}{-2}$$

n=\frac{-21±9}{-2} tenglamasini yeching, bunda ± musbat. -21 ni 9 ga qo'shish.

$$n=-\frac{12}{-2}$$

-12 ni -2 ga bo'lish.

$$n=6$$

n=\frac{-21±9}{-2} tenglamasini yeching, bunda ± manfiy. -21 dan 9 ni ayirish.

$$n=-\frac{30}{-2}$$

-30 ni -2 ga bo'lish.

$$n=15$$

Tenglama yechildi.

$$n=6 n=15$$

Kvadratni yakunlash bo'yicha qadamlar

12 ga n-4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$12n-48-30=n^{2}-9n+12$$

-78 olish uchun -48 dan 30 ni ayirish.

$$12n-78=n^{2}-9n+12$$

Ikkala tarafdan n^{2} ni ayirish.

$$12n-78-n^{2}=-9n+12$$

9n ni ikki tarafga qo’shing.

$$12n-78-n^{2}+9n=12$$

21n ni olish uchun 12n va 9n ni birlashtirish.

$$21n-78-n^{2}=12$$

78 ni ikki tarafga qo’shing.

$$21n-n^{2}=12+78$$

90 olish uchun 12 va 78'ni qo'shing.

$$21n-n^{2}=90$$

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

$$-n^{2}+21n=90$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-n^{2}+21n}{-1}=\frac{90}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$n^{2}+\frac{21}{-1}n=\frac{90}{-1}$$

21 ni -1 ga bo'lish.

$$n^{2}-21n=\frac{90}{-1}$$

90 ni -1 ga bo'lish.

$$n^{2}-21n=-90$$

-21 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{21}{2} olish uchun. Keyin, -\frac{21}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$n^{2}-21n+\left(-\frac{21}{2}\right)^{2}=-90+\left(-\frac{21}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{21}{2} kvadratini chiqarish.

$$n^{2}-21n+\frac{441}{4}=-90+\frac{441}{4}$$

-90 ni \frac{441}{4} ga qo'shish.

$$n^{2}-21n+\frac{441}{4}=\frac{81}{4}$$

n^{2}-21n+\frac{441}{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.

$$\left(n-\frac{21}{2}\right)^{2}=\frac{81}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(n-\frac{21}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Qisqartirish.

$$n-\frac{21}{2}=\frac{9}{2} n-\frac{21}{2}=-\frac{9}{2}$$

\frac{21}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$n=15 n=6$$