14-(5x-1)(2x+3)=17-(10x+19(x-6) eching

$$14 - ( 5 x - 1 ) ( 2 x + 3 ) = 17 - ( 10 x + 19 ( x - 6 )$$

x uchun yechish (complex solution)

$$x=\frac{4+\sqrt{269}i}{5}\approx 0,8+3,280243893i$$

$$x=\frac{-\sqrt{269}i+4}{5}\approx 0,8-3,280243893i$$

Kvadrat tenglamalaridan foydalanish qoidalari

5x-1 ga 2x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$14-\left(10x^{2}+13x-3\right)=17-\left(10x+19\left(x-6\right)\right)$$

10x^{2}+13x-3 teskarisini topish uchun har birining teskarisini toping.

$$14-10x^{2}-13x+3=17-\left(10x+19\left(x-6\right)\right)$$

17 olish uchun 14 va 3'ni qo'shing.

$$17-10x^{2}-13x=17-\left(10x+19\left(x-6\right)\right)$$

19 ga x-6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$17-10x^{2}-13x=17-\left(10x+19x-114\right)$$

29x ni olish uchun 10x va 19x ni birlashtirish.

$$17-10x^{2}-13x=17-\left(29x-114\right)$$

29x-114 teskarisini topish uchun har birining teskarisini toping.

$$17-10x^{2}-13x=17-29x+114$$

131 olish uchun 17 va 114'ni qo'shing.

$$17-10x^{2}-13x=131-29x$$

Ikkala tarafdan 131 ni ayirish.

$$17-10x^{2}-13x-131=-29x$$

-114 olish uchun 17 dan 131 ni ayirish.

$$-114-10x^{2}-13x=-29x$$

29x ni ikki tarafga qo’shing.

$$-114-10x^{2}-13x+29x=0$$

16x ni olish uchun -13x va 29x ni birlashtirish.

$$-114-10x^{2}+16x=0$$

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.

$$-10x^{2}+16x-114=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -10 ni a, 16 ni b va -114 ni c bilan almashtiring.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-10\right)\left(-114\right)}}{2\left(-10\right)}$$

16 kvadratini chiqarish.

$$x=\frac{-16±\sqrt{256-4\left(-10\right)\left(-114\right)}}{2\left(-10\right)}$$

-4 ni -10 marotabaga ko'paytirish.

$$x=\frac{-16±\sqrt{256+40\left(-114\right)}}{2\left(-10\right)}$$

40 ni -114 marotabaga ko'paytirish.

$$x=\frac{-16±\sqrt{256-4560}}{2\left(-10\right)}$$

256 ni -4560 ga qo'shish.

$$x=\frac{-16±\sqrt{-4304}}{2\left(-10\right)}$$

-4304 ning kvadrat ildizini chiqarish.

$$x=\frac{-16±4\sqrt{269}i}{2\left(-10\right)}$$

2 ni -10 marotabaga ko'paytirish.

$$x=\frac{-16±4\sqrt{269}i}{-20}$$

x=\frac{-16±4\sqrt{269}i}{-20} tenglamasini yeching, bunda ± musbat. -16 ni 4i\sqrt{269} ga qo'shish.

$$x=\frac{-16+4\sqrt{269}i}{-20}$$

-16+4i\sqrt{269} ni -20 ga bo'lish.

$$x=\frac{-\sqrt{269}i+4}{5}$$

x=\frac{-16±4\sqrt{269}i}{-20} tenglamasini yeching, bunda ± manfiy. -16 dan 4i\sqrt{269} ni ayirish.

$$x=\frac{-4\sqrt{269}i-16}{-20}$$

-16-4i\sqrt{269} ni -20 ga bo'lish.

$$x=\frac{4+\sqrt{269}i}{5}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{269}i+4}{5} x=\frac{4+\sqrt{269}i}{5}$$

Kvadratni yakunlash bo'yicha qadamlar

5x-1 ga 2x+3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$14-\left(10x^{2}+13x-3\right)=17-\left(10x+19\left(x-6\right)\right)$$

10x^{2}+13x-3 teskarisini topish uchun har birining teskarisini toping.

$$14-10x^{2}-13x+3=17-\left(10x+19\left(x-6\right)\right)$$

17 olish uchun 14 va 3'ni qo'shing.

$$17-10x^{2}-13x=17-\left(10x+19\left(x-6\right)\right)$$

19 ga x-6 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$17-10x^{2}-13x=17-\left(10x+19x-114\right)$$

29x ni olish uchun 10x va 19x ni birlashtirish.

$$17-10x^{2}-13x=17-\left(29x-114\right)$$

29x-114 teskarisini topish uchun har birining teskarisini toping.

$$17-10x^{2}-13x=17-29x+114$$

131 olish uchun 17 va 114'ni qo'shing.

$$17-10x^{2}-13x=131-29x$$

29x ni ikki tarafga qo’shing.

$$17-10x^{2}-13x+29x=131$$

16x ni olish uchun -13x va 29x ni birlashtirish.

$$17-10x^{2}+16x=131$$

Ikkala tarafdan 17 ni ayirish.

$$-10x^{2}+16x=131-17$$

114 olish uchun 131 dan 17 ni ayirish.

$$-10x^{2}+16x=114$$

Ikki tarafini -10 ga bo‘ling.

$$\frac{-10x^{2}+16x}{-10}=\frac{114}{-10}$$

-10 ga bo'lish -10 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{16}{-10}x=\frac{114}{-10}$$

\frac{16}{-10} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{8}{5}x=\frac{114}{-10}$$

\frac{114}{-10} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{8}{5}x=-\frac{57}{5}$$

-\frac{8}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{4}{5} olish uchun. Keyin, -\frac{4}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{8}{5}x+\left(-\frac{4}{5}\right)^{2}=-\frac{57}{5}+\left(-\frac{4}{5}\right)^{2}$$

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

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{57}{5}+\frac{16}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{57}{5} ni \frac{16}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{8}{5}x+\frac{16}{25}=-\frac{269}{25}$$

x^{2}-\frac{8}{5}x+\frac{16}{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.

$$\left(x-\frac{4}{5}\right)^{2}=-\frac{269}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{4}{5}\right)^{2}}=\sqrt{-\frac{269}{25}}$$

Qisqartirish.

$$x-\frac{4}{5}=\frac{\sqrt{269}i}{5} x-\frac{4}{5}=-\frac{\sqrt{269}i}{5}$$

\frac{4}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{4+\sqrt{269}i}{5} x=\frac{-\sqrt{269}i+4}{5}$$