5w^2=-16w-3 eching

$$5 w ^ { 2 } = - 16 w - 3$$

w uchun yechish

$$w=-3$$

$$w=-\frac{1}{5}=-0,2$$

Guruhlash orqali faktorlash qadamlari

16w ni ikki tarafga qo’shing.

$$5w^{2}+16w=-3$$

3 ni ikki tarafga qo’shing.

$$5w^{2}+16w+3=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 5w^{2}+aw+bw+3 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=16 ab=5\times 3=15$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 15-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,15 3,5$$

Har bir juftlik yigʻindisini hisoblang.

$$1+15=16 3+5=8$$

Yechim – 16 yigʻindisini beruvchi juftlik.

$$a=1 b=15$$

5w^{2}+16w+3 ni \left(5w^{2}+w\right)+\left(15w+3\right) sifatida qaytadan yozish.

$$\left(5w^{2}+w\right)+\left(15w+3\right)$$

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

$$w\left(5w+1\right)+3\left(5w+1\right)$$

Distributiv funktsiyasidan foydalangan holda 5w+1 umumiy terminini chiqaring.

$$\left(5w+1\right)\left(w+3\right)$$

Tenglamani yechish uchun 5w+1=0 va w+3=0 ni yeching.

$$w=-\frac{1}{5} w=-3$$

Kvadrat tenglamalaridan foydalanish qoidalari

16w ni ikki tarafga qo’shing.

$$5w^{2}+16w=-3$$

3 ni ikki tarafga qo’shing.

$$5w^{2}+16w+3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, 16 ni b va 3 ni c bilan almashtiring.

$$w=\frac{-16±\sqrt{16^{2}-4\times 5\times 3}}{2\times 5}$$

16 kvadratini chiqarish.

$$w=\frac{-16±\sqrt{256-4\times 5\times 3}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$w=\frac{-16±\sqrt{256-20\times 3}}{2\times 5}$$

-20 ni 3 marotabaga ko'paytirish.

$$w=\frac{-16±\sqrt{256-60}}{2\times 5}$$

256 ni -60 ga qo'shish.

$$w=\frac{-16±\sqrt{196}}{2\times 5}$$

196 ning kvadrat ildizini chiqarish.

$$w=\frac{-16±14}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$w=\frac{-16±14}{10}$$

w=\frac{-16±14}{10} tenglamasini yeching, bunda ± musbat. -16 ni 14 ga qo'shish.

$$w=-\frac{2}{10}$$

\frac{-2}{10} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$w=-\frac{1}{5}$$

w=\frac{-16±14}{10} tenglamasini yeching, bunda ± manfiy. -16 dan 14 ni ayirish.

$$w=-\frac{30}{10}$$

-30 ni 10 ga bo'lish.

$$w=-3$$

Tenglama yechildi.

$$w=-\frac{1}{5} w=-3$$

Kvadratni yakunlash bo'yicha qadamlar

16w ni ikki tarafga qo’shing.

$$5w^{2}+16w=-3$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5w^{2}+16w}{5}=-\frac{3}{5}$$

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

$$w^{2}+\frac{16}{5}w=-\frac{3}{5}$$

\frac{16}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{8}{5} olish uchun. Keyin, \frac{8}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$w^{2}+\frac{16}{5}w+\left(\frac{8}{5}\right)^{2}=-\frac{3}{5}+\left(\frac{8}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{8}{5} kvadratini chiqarish.

$$w^{2}+\frac{16}{5}w+\frac{64}{25}=-\frac{3}{5}+\frac{64}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{5} ni \frac{64}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$w^{2}+\frac{16}{5}w+\frac{64}{25}=\frac{49}{25}$$

w^{2}+\frac{16}{5}w+\frac{64}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(w+\frac{8}{5}\right)^{2}=\frac{49}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(w+\frac{8}{5}\right)^{2}}=\sqrt{\frac{49}{25}}$$

Qisqartirish.

$$w+\frac{8}{5}=\frac{7}{5} w+\frac{8}{5}=-\frac{7}{5}$$

Tenglamaning ikkala tarafidan \frac{8}{5} ni ayirish.

$$w=-\frac{1}{5} w=-3$$