t^2+3t/2=t+7/4 eching

$$\frac { t ^ { 2 } + 3 t } { 2 } = \frac { t + 7 } { 4 }$$

t uchun yechish

$$t = -\frac{7}{2} = -3\frac{1}{2} = -3,5$$

$$t=1$$

Guruhlash orqali faktorlash qadamlari

Tenglamaning ikkala tarafini 4 ga, 2,4 ning eng kichik karralisiga ko‘paytiring.

$$2\left(t^{2}+3t\right)=t+7$$

2 ga t^{2}+3t ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2t^{2}+6t=t+7$$

Ikkala tarafdan t ni ayirish.

$$2t^{2}+6t-t=7$$

5t ni olish uchun 6t va -t ni birlashtirish.

$$2t^{2}+5t=7$$

Ikkala tarafdan 7 ni ayirish.

$$2t^{2}+5t-7=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 2t^{2}+at+bt-7 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=5 ab=2\left(-7\right)=-14$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -14-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,14 -2,7$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+14=13 -2+7=5$$

Yechim – 5 yigʻindisini beruvchi juftlik.

$$a=-2 b=7$$

2t^{2}+5t-7 ni \left(2t^{2}-2t\right)+\left(7t-7\right) sifatida qaytadan yozish.

$$\left(2t^{2}-2t\right)+\left(7t-7\right)$$

Birinchi guruhda 2t ni va ikkinchi guruhda 7 ni faktordan chiqaring.

$$2t\left(t-1\right)+7\left(t-1\right)$$

Distributiv funktsiyasidan foydalangan holda t-1 umumiy terminini chiqaring.

$$\left(t-1\right)\left(2t+7\right)$$

Tenglamani yechish uchun t-1=0 va 2t+7=0 ni yeching.

$$t=1 t=-\frac{7}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 4 ga, 2,4 ning eng kichik karralisiga ko‘paytiring.

$$2\left(t^{2}+3t\right)=t+7$$

2 ga t^{2}+3t ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2t^{2}+6t=t+7$$

Ikkala tarafdan t ni ayirish.

$$2t^{2}+6t-t=7$$

5t ni olish uchun 6t va -t ni birlashtirish.

$$2t^{2}+5t=7$$

Ikkala tarafdan 7 ni ayirish.

$$2t^{2}+5t-7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 5 ni b va -7 ni c bilan almashtiring.

$$t=\frac{-5±\sqrt{5^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

5 kvadratini chiqarish.

$$t=\frac{-5±\sqrt{25-4\times 2\left(-7\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$t=\frac{-5±\sqrt{25-8\left(-7\right)}}{2\times 2}$$

-8 ni -7 marotabaga ko'paytirish.

$$t=\frac{-5±\sqrt{25+56}}{2\times 2}$$

25 ni 56 ga qo'shish.

$$t=\frac{-5±\sqrt{81}}{2\times 2}$$

81 ning kvadrat ildizini chiqarish.

$$t=\frac{-5±9}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$t=\frac{-5±9}{4}$$

t=\frac{-5±9}{4} tenglamasini yeching, bunda ± musbat. -5 ni 9 ga qo'shish.

$$t=\frac{4}{4}$$

4 ni 4 ga bo'lish.

$$t=1$$

t=\frac{-5±9}{4} tenglamasini yeching, bunda ± manfiy. -5 dan 9 ni ayirish.

$$t=-\frac{14}{4}$$

\frac{-14}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$t=-\frac{7}{2}$$

Tenglama yechildi.

$$t=1 t=-\frac{7}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 4 ga, 2,4 ning eng kichik karralisiga ko‘paytiring.

$$2\left(t^{2}+3t\right)=t+7$$

2 ga t^{2}+3t ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2t^{2}+6t=t+7$$

Ikkala tarafdan t ni ayirish.

$$2t^{2}+6t-t=7$$

5t ni olish uchun 6t va -t ni birlashtirish.

$$2t^{2}+5t=7$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2t^{2}+5t}{2}=\frac{7}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$t^{2}+\frac{5}{2}t=\frac{7}{2}$$

\frac{5}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{4} olish uchun. Keyin, \frac{5}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}+\frac{5}{2}t+\left(\frac{5}{4}\right)^{2}=\frac{7}{2}+\left(\frac{5}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{4} kvadratini chiqarish.

$$t^{2}+\frac{5}{2}t+\frac{25}{16}=\frac{7}{2}+\frac{25}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{7}{2} ni \frac{25}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}+\frac{5}{2}t+\frac{25}{16}=\frac{81}{16}$$

t^{2}+\frac{5}{2}t+\frac{25}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t+\frac{5}{4}\right)^{2}=\frac{81}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t+\frac{5}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Qisqartirish.

$$t+\frac{5}{4}=\frac{9}{4} t+\frac{5}{4}=-\frac{9}{4}$$

Tenglamaning ikkala tarafidan \frac{5}{4} ni ayirish.

$$t=1 t=-\frac{7}{2}$$