2x^2=7x-3 eching

$$2 x ^ { 2 } = 7 x - 3$$

x uchun yechish

$$x=\frac{1}{2}=0,5$$

$$x=3$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 7x ni ayirish.

$$2x^{2}-7x=-3$$

3 ni ikki tarafga qo’shing.

$$2x^{2}-7x+3=0$$

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

$$a+b=-7 ab=2\times 3=6$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 6-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-6 -2,-3$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-6=-7 -2-3=-5$$

Yechim – -7 yigʻindisini beruvchi juftlik.

$$a=-6 b=-1$$

2x^{2}-7x+3 ni \left(2x^{2}-6x\right)+\left(-x+3\right) sifatida qaytadan yozish.

$$\left(2x^{2}-6x\right)+\left(-x+3\right)$$

Birinchi guruhda 2x ni va ikkinchi guruhda -1 ni faktordan chiqaring.

$$2x\left(x-3\right)-\left(x-3\right)$$

Distributiv funktsiyasidan foydalangan holda x-3 umumiy terminini chiqaring.

$$\left(x-3\right)\left(2x-1\right)$$

Tenglamani yechish uchun x-3=0 va 2x-1=0 ni yeching.

$$x=3 x=\frac{1}{2}$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 7x ni ayirish.

$$2x^{2}-7x=-3$$

3 ni ikki tarafga qo’shing.

$$2x^{2}-7x+3=0$$

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 3 ni c bilan almashtiring.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 2\times 3}}{2\times 2}$$

-7 kvadratini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 2\times 3}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-8\times 3}}{2\times 2}$$

-8 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-24}}{2\times 2}$$

49 ni -24 ga qo'shish.

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

25 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-7\right)±5}{2\times 2}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±5}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{7±5}{4}$$

x=\frac{7±5}{4} tenglamasini yeching, bunda ± musbat. 7 ni 5 ga qo'shish.

$$x=\frac{12}{4}$$

12 ni 4 ga bo'lish.

$$x=3$$

x=\frac{7±5}{4} tenglamasini yeching, bunda ± manfiy. 7 dan 5 ni ayirish.

$$x=\frac{2}{4}$$

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

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

Tenglama yechildi.

$$x=3 x=\frac{1}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 7x ni ayirish.

$$2x^{2}-7x=-3$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-7x}{2}=-\frac{3}{2}$$

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

$$x^{2}-\frac{7}{2}x=-\frac{3}{2}$$

-\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.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=-\frac{3}{2}+\left(-\frac{7}{4}\right)^{2}$$

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

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-\frac{3}{2}+\frac{49}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{2} ni \frac{49}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{25}{16}$$

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.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{25}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Qisqartirish.

$$x-\frac{7}{4}=\frac{5}{4} x-\frac{7}{4}=-\frac{5}{4}$$

\frac{7}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=3 x=\frac{1}{2}$$