20p^2+33p+16=6 eching

$$20 p ^ { 2 } + 33 p + 16 = 6$$

p uchun yechish

$$p = -\frac{5}{4} = -1\frac{1}{4} = -1,25$$

$$p=-\frac{2}{5}=-0,4$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 6 ni ayirish.

$$20p^{2}+33p+16-6=0$$

10 olish uchun 16 dan 6 ni ayirish.

$$20p^{2}+33p+10=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 20p^{2}+ap+bp+10 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=33 ab=20\times 10=200$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 200-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,200 2,100 4,50 5,40 8,25 10,20$$

Har bir juftlik yigʻindisini hisoblang.

$$1+200=201 2+100=102 4+50=54 5+40=45 8+25=33 10+20=30$$

Yechim – 33 yigʻindisini beruvchi juftlik.

$$a=8 b=25$$

20p^{2}+33p+10 ni \left(20p^{2}+8p\right)+\left(25p+10\right) sifatida qaytadan yozish.

$$\left(20p^{2}+8p\right)+\left(25p+10\right)$$

Birinchi guruhda 4p ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$4p\left(5p+2\right)+5\left(5p+2\right)$$

Distributiv funktsiyasidan foydalangan holda 5p+2 umumiy terminini chiqaring.

$$\left(5p+2\right)\left(4p+5\right)$$

Tenglamani yechish uchun 5p+2=0 va 4p+5=0 ni yeching.

$$p=-\frac{2}{5} p=-\frac{5}{4}$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$20p^{2}+33p+16=6$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$20p^{2}+33p+16-6=6-6$$

O‘zidan 6 ayirilsa 0 qoladi.

$$20p^{2}+33p+16-6=0$$

16 dan 6 ni ayirish.

$$20p^{2}+33p+10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 20 ni a, 33 ni b va 10 ni c bilan almashtiring.

$$p=\frac{-33±\sqrt{33^{2}-4\times 20\times 10}}{2\times 20}$$

33 kvadratini chiqarish.

$$p=\frac{-33±\sqrt{1089-4\times 20\times 10}}{2\times 20}$$

-4 ni 20 marotabaga ko'paytirish.

$$p=\frac{-33±\sqrt{1089-80\times 10}}{2\times 20}$$

-80 ni 10 marotabaga ko'paytirish.

$$p=\frac{-33±\sqrt{1089-800}}{2\times 20}$$

1089 ni -800 ga qo'shish.

$$p=\frac{-33±\sqrt{289}}{2\times 20}$$

289 ning kvadrat ildizini chiqarish.

$$p=\frac{-33±17}{2\times 20}$$

2 ni 20 marotabaga ko'paytirish.

$$p=\frac{-33±17}{40}$$

p=\frac{-33±17}{40} tenglamasini yeching, bunda ± musbat. -33 ni 17 ga qo'shish.

$$p=-\frac{16}{40}$$

\frac{-16}{40} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$p=-\frac{2}{5}$$

p=\frac{-33±17}{40} tenglamasini yeching, bunda ± manfiy. -33 dan 17 ni ayirish.

$$p=-\frac{50}{40}$$

\frac{-50}{40} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$p=-\frac{5}{4}$$

Tenglama yechildi.

$$p=-\frac{2}{5} p=-\frac{5}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$20p^{2}+33p+16=6$$

Tenglamaning ikkala tarafidan 16 ni ayirish.

$$20p^{2}+33p+16-16=6-16$$

O‘zidan 16 ayirilsa 0 qoladi.

$$20p^{2}+33p=6-16$$

6 dan 16 ni ayirish.

$$20p^{2}+33p=-10$$

Ikki tarafini 20 ga bo‘ling.

$$\frac{20p^{2}+33p}{20}=-\frac{10}{20}$$

20 ga bo'lish 20 ga ko'paytirishni bekor qiladi.

$$p^{2}+\frac{33}{20}p=-\frac{10}{20}$$

\frac{-10}{20} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$p^{2}+\frac{33}{20}p=-\frac{1}{2}$$

\frac{33}{20} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{33}{40} olish uchun. Keyin, \frac{33}{40} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$p^{2}+\frac{33}{20}p+\left(\frac{33}{40}\right)^{2}=-\frac{1}{2}+\left(\frac{33}{40}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{33}{40} kvadratini chiqarish.

$$p^{2}+\frac{33}{20}p+\frac{1089}{1600}=-\frac{1}{2}+\frac{1089}{1600}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{2} ni \frac{1089}{1600} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$p^{2}+\frac{33}{20}p+\frac{1089}{1600}=\frac{289}{1600}$$

p^{2}+\frac{33}{20}p+\frac{1089}{1600} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(p+\frac{33}{40}\right)^{2}=\frac{289}{1600}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(p+\frac{33}{40}\right)^{2}}=\sqrt{\frac{289}{1600}}$$

Qisqartirish.

$$p+\frac{33}{40}=\frac{17}{40} p+\frac{33}{40}=-\frac{17}{40}$$

Tenglamaning ikkala tarafidan \frac{33}{40} ni ayirish.

$$p=-\frac{2}{5} p=-\frac{5}{4}$$