2x^2+15x=8x-5 eching

$$2 x ^ { 2 } + 15 x = 8 x - 5$$

x uchun yechish

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2,5$$

$$x=-1$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 8x ni ayirish.

$$2x^{2}+15x-8x=-5$$

7x ni olish uchun 15x va -8x ni birlashtirish.

$$2x^{2}+7x=-5$$

5 ni ikki tarafga qo’shing.

$$2x^{2}+7x+5=0$$

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

$$a+b=7 ab=2\times 5=10$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 10-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,10 2,5$$

Har bir juftlik yigʻindisini hisoblang.

$$1+10=11 2+5=7$$

Yechim – 7 yigʻindisini beruvchi juftlik.

$$a=2 b=5$$

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

$$\left(2x^{2}+2x\right)+\left(5x+5\right)$$

Birinchi guruhda 2x ni va ikkinchi guruhda 5 ni faktordan chiqaring.

$$2x\left(x+1\right)+5\left(x+1\right)$$

Distributiv funktsiyasidan foydalangan holda x+1 umumiy terminini chiqaring.

$$\left(x+1\right)\left(2x+5\right)$$

Tenglamani yechish uchun x+1=0 va 2x+5=0 ni yeching.

$$x=-1 x=-\frac{5}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 8x ni ayirish.

$$2x^{2}+15x-8x=-5$$

7x ni olish uchun 15x va -8x ni birlashtirish.

$$2x^{2}+7x=-5$$

5 ni ikki tarafga qo’shing.

$$2x^{2}+7x+5=0$$

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

$$x=\frac{-7±\sqrt{7^{2}-4\times 2\times 5}}{2\times 2}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\times 2\times 5}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-8\times 5}}{2\times 2}$$

-8 ni 5 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-40}}{2\times 2}$$

49 ni -40 ga qo'shish.

$$x=\frac{-7±\sqrt{9}}{2\times 2}$$

9 ning kvadrat ildizini chiqarish.

$$x=\frac{-7±3}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-7±3}{4}$$

x=\frac{-7±3}{4} tenglamasini yeching, bunda ± musbat. -7 ni 3 ga qo'shish.

$$x=-\frac{4}{4}$$

-4 ni 4 ga bo'lish.

$$x=-1$$

x=\frac{-7±3}{4} tenglamasini yeching, bunda ± manfiy. -7 dan 3 ni ayirish.

$$x=-\frac{10}{4}$$

\frac{-10}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Tenglama yechildi.

$$x=-1 x=-\frac{5}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 8x ni ayirish.

$$2x^{2}+15x-8x=-5$$

7x ni olish uchun 15x va -8x ni birlashtirish.

$$2x^{2}+7x=-5$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+7x}{2}=-\frac{5}{2}$$

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

$$x^{2}+\frac{7}{2}x=-\frac{5}{2}$$

\frac{7}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{4} olish uchun. Keyin, \frac{7}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{4} kvadratini chiqarish.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=-\frac{5}{2}+\frac{49}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{2} ni \frac{49}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{9}{16}$$

x^{2}+\frac{7}{2}x+\frac{49}{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.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{9}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Qisqartirish.

$$x+\frac{7}{4}=\frac{3}{4} x+\frac{7}{4}=-\frac{3}{4}$$

Tenglamaning ikkala tarafidan \frac{7}{4} ni ayirish.

$$x=-1 x=-\frac{5}{2}$$