16a^2+21a+9=6a^2 eching

$$16 a ^ { 2 } + 21 a + 9 = 6 a ^ { 2 }$$

a uchun yechish

$$a=-\frac{3}{5}=-0,6$$

$$a = -\frac{3}{2} = -1\frac{1}{2} = -1,5$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 6a^{2} ni ayirish.

$$16a^{2}+21a+9-6a^{2}=0$$

10a^{2} ni olish uchun 16a^{2} va -6a^{2} ni birlashtirish.

$$10a^{2}+21a+9=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 10a^{2}+aa+ba+9 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=21 ab=10\times 9=90$$

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.

$$a=6 b=15$$

10a^{2}+21a+9 ni \left(10a^{2}+6a\right)+\left(15a+9\right) sifatida qaytadan yozish.

$$\left(10a^{2}+6a\right)+\left(15a+9\right)$$

Birinchi guruhda 2a ni va ikkinchi guruhda 3 ni faktordan chiqaring.

$$2a\left(5a+3\right)+3\left(5a+3\right)$$

Distributiv funktsiyasidan foydalangan holda 5a+3 umumiy terminini chiqaring.

$$\left(5a+3\right)\left(2a+3\right)$$

Tenglamani yechish uchun 5a+3=0 va 2a+3=0 ni yeching.

$$a=-\frac{3}{5} a=-\frac{3}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 6a^{2} ni ayirish.

$$16a^{2}+21a+9-6a^{2}=0$$

10a^{2} ni olish uchun 16a^{2} va -6a^{2} ni birlashtirish.

$$10a^{2}+21a+9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 10 ni a, 21 ni b va 9 ni c bilan almashtiring.

$$a=\frac{-21±\sqrt{21^{2}-4\times 10\times 9}}{2\times 10}$$

21 kvadratini chiqarish.

$$a=\frac{-21±\sqrt{441-4\times 10\times 9}}{2\times 10}$$

-4 ni 10 marotabaga ko'paytirish.

$$a=\frac{-21±\sqrt{441-40\times 9}}{2\times 10}$$

-40 ni 9 marotabaga ko'paytirish.

$$a=\frac{-21±\sqrt{441-360}}{2\times 10}$$

441 ni -360 ga qo'shish.

$$a=\frac{-21±\sqrt{81}}{2\times 10}$$

81 ning kvadrat ildizini chiqarish.

$$a=\frac{-21±9}{2\times 10}$$

2 ni 10 marotabaga ko'paytirish.

$$a=\frac{-21±9}{20}$$

a=\frac{-21±9}{20} tenglamasini yeching, bunda ± musbat. -21 ni 9 ga qo'shish.

$$a=-\frac{12}{20}$$

\frac{-12}{20} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$a=-\frac{3}{5}$$

a=\frac{-21±9}{20} tenglamasini yeching, bunda ± manfiy. -21 dan 9 ni ayirish.

$$a=-\frac{30}{20}$$

\frac{-30}{20} ulushini 10 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$a=-\frac{3}{2}$$

Tenglama yechildi.

$$a=-\frac{3}{5} a=-\frac{3}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 6a^{2} ni ayirish.

$$16a^{2}+21a+9-6a^{2}=0$$

10a^{2} ni olish uchun 16a^{2} va -6a^{2} ni birlashtirish.

$$10a^{2}+21a+9=0$$

Ikkala tarafdan 9 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$10a^{2}+21a=-9$$

Ikki tarafini 10 ga bo‘ling.

$$\frac{10a^{2}+21a}{10}=-\frac{9}{10}$$

10 ga bo'lish 10 ga ko'paytirishni bekor qiladi.

$$a^{2}+\frac{21}{10}a=-\frac{9}{10}$$

\frac{21}{10} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{21}{20} olish uchun. Keyin, \frac{21}{20} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$a^{2}+\frac{21}{10}a+\left(\frac{21}{20}\right)^{2}=-\frac{9}{10}+\left(\frac{21}{20}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{21}{20} kvadratini chiqarish.

$$a^{2}+\frac{21}{10}a+\frac{441}{400}=-\frac{9}{10}+\frac{441}{400}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{9}{10} ni \frac{441}{400} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$a^{2}+\frac{21}{10}a+\frac{441}{400}=\frac{81}{400}$$

a^{2}+\frac{21}{10}a+\frac{441}{400} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(a+\frac{21}{20}\right)^{2}=\frac{81}{400}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(a+\frac{21}{20}\right)^{2}}=\sqrt{\frac{81}{400}}$$

Qisqartirish.

$$a+\frac{21}{20}=\frac{9}{20} a+\frac{21}{20}=-\frac{9}{20}$$

Tenglamaning ikkala tarafidan \frac{21}{20} ni ayirish.

$$a=-\frac{3}{5} a=-\frac{3}{2}$$