5x-2(x-1)(3-x)=11 eching

$$5 x - 2 ( x - 1 ) ( 3 - x ) = 11$$

x uchun yechish

$$x=-1$$

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

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 11 ni ayirish.

$$5x-2\left(x-1\right)\left(3-x\right)-11=0$$

-2 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x+\left(-2x+2\right)\left(3-x\right)-11=0$$

-2x+2 ga 3-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$5x-8x+2x^{2}+6-11=0$$

-3x ni olish uchun 5x va -8x ni birlashtirish.

$$-3x+2x^{2}+6-11=0$$

-5 olish uchun 6 dan 11 ni ayirish.

$$-3x+2x^{2}-5=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$2x^{2}-3x-5=0$$

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

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-5\right)}}{2\times 2}$$

-3 kvadratini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-5\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\left(-5\right)}}{2\times 2}$$

-8 ni -5 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9+40}}{2\times 2}$$

9 ni 40 ga qo'shish.

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

49 ning kvadrat ildizini chiqarish.

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

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±7}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{3±7}{4}$$

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

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

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

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

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

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

-4 ni 4 ga bo'lish.

$$x=-1$$

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

-2 hosil qilish uchun -1 va 2 ni ko'paytirish.

$$5x-2\left(x-1\right)\left(3-x\right)=11$$

-2 ga x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$5x+\left(-2x+2\right)\left(3-x\right)=11$$

-2x+2 ga 3-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$5x-8x+2x^{2}+6=11$$

-3x ni olish uchun 5x va -8x ni birlashtirish.

$$-3x+2x^{2}+6=11$$

Ikkala tarafdan 6 ni ayirish.

$$-3x+2x^{2}=11-6$$

5 olish uchun 11 dan 6 ni ayirish.

$$-3x+2x^{2}=5$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$2x^{2}-3x=5$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-3x}{2}=\frac{5}{2}$$

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

$$x^{2}-\frac{3}{2}x=\frac{5}{2}$$

-\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{4} olish uchun. Keyin, -\frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\frac{5}{2}+\left(-\frac{3}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{5}{2}+\frac{9}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{2} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{49}{16}$$

x^{2}-\frac{3}{2}x+\frac{9}{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.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{49}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Qisqartirish.

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

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

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