3*(2x-10)2(3x-30)=-5(3x+100) eching

$$3 \times (2x-10)2(3x-30)=-5(3x+100)$$

x uchun yechish (complex solution)

$$x=\frac{175+5\sqrt{247}i}{24}\approx 7,291666667+3,274215343i$$

$$x=\frac{-5\sqrt{247}i+175}{24}\approx 7,291666667-3,274215343i$$

Kvadrat tenglamalaridan foydalanish qoidalari

6 hosil qilish uchun 3 va 2 ni ko'paytirish.

$$6\left(2x-10\right)\left(3x-30\right)=-5\left(3x+100\right)$$

6 ga 2x-10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(12x-60\right)\left(3x-30\right)=-5\left(3x+100\right)$$

12x-60 ga 3x-30 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$36x^{2}-540x+1800=-5\left(3x+100\right)$$

-5 ga 3x+100 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x^{2}-540x+1800=-15x-500$$

15x ni ikki tarafga qo’shing.

$$36x^{2}-540x+1800+15x=-500$$

-525x ni olish uchun -540x va 15x ni birlashtirish.

$$36x^{2}-525x+1800=-500$$

500 ni ikki tarafga qo’shing.

$$36x^{2}-525x+1800+500=0$$

2300 olish uchun 1800 va 500'ni qo'shing.

$$36x^{2}-525x+2300=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 36 ni a, -525 ni b va 2300 ni c bilan almashtiring.

$$x=\frac{-\left(-525\right)±\sqrt{\left(-525\right)^{2}-4\times 36\times 2300}}{2\times 36}$$

-525 kvadratini chiqarish.

$$x=\frac{-\left(-525\right)±\sqrt{275625-4\times 36\times 2300}}{2\times 36}$$

-4 ni 36 marotabaga ko'paytirish.

$$x=\frac{-\left(-525\right)±\sqrt{275625-144\times 2300}}{2\times 36}$$

-144 ni 2300 marotabaga ko'paytirish.

$$x=\frac{-\left(-525\right)±\sqrt{275625-331200}}{2\times 36}$$

275625 ni -331200 ga qo'shish.

$$x=\frac{-\left(-525\right)±\sqrt{-55575}}{2\times 36}$$

-55575 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-525\right)±15\sqrt{247}i}{2\times 36}$$

-525 ning teskarisi 525 ga teng.

$$x=\frac{525±15\sqrt{247}i}{2\times 36}$$

2 ni 36 marotabaga ko'paytirish.

$$x=\frac{525±15\sqrt{247}i}{72}$$

x=\frac{525±15\sqrt{247}i}{72} tenglamasini yeching, bunda ± musbat. 525 ni 15i\sqrt{247} ga qo'shish.

$$x=\frac{525+15\sqrt{247}i}{72}$$

525+15i\sqrt{247} ni 72 ga bo'lish.

$$x=\frac{175+5\sqrt{247}i}{24}$$

x=\frac{525±15\sqrt{247}i}{72} tenglamasini yeching, bunda ± manfiy. 525 dan 15i\sqrt{247} ni ayirish.

$$x=\frac{-15\sqrt{247}i+525}{72}$$

525-15i\sqrt{247} ni 72 ga bo'lish.

$$x=\frac{-5\sqrt{247}i+175}{24}$$

Tenglama yechildi.

$$x=\frac{175+5\sqrt{247}i}{24} x=\frac{-5\sqrt{247}i+175}{24}$$

Kvadratni yakunlash bo'yicha qadamlar

6 hosil qilish uchun 3 va 2 ni ko'paytirish.

$$6\left(2x-10\right)\left(3x-30\right)=-5\left(3x+100\right)$$

6 ga 2x-10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\left(12x-60\right)\left(3x-30\right)=-5\left(3x+100\right)$$

12x-60 ga 3x-30 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$36x^{2}-540x+1800=-5\left(3x+100\right)$$

-5 ga 3x+100 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$36x^{2}-540x+1800=-15x-500$$

15x ni ikki tarafga qo’shing.

$$36x^{2}-540x+1800+15x=-500$$

-525x ni olish uchun -540x va 15x ni birlashtirish.

$$36x^{2}-525x+1800=-500$$

Ikkala tarafdan 1800 ni ayirish.

$$36x^{2}-525x=-500-1800$$

-2300 olish uchun -500 dan 1800 ni ayirish.

$$36x^{2}-525x=-2300$$

Ikki tarafini 36 ga bo‘ling.

$$\frac{36x^{2}-525x}{36}=-\frac{2300}{36}$$

36 ga bo'lish 36 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{525}{36}\right)x=-\frac{2300}{36}$$

\frac{-525}{36} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{175}{12}x=-\frac{2300}{36}$$

\frac{-2300}{36} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{175}{12}x=-\frac{575}{9}$$

-\frac{175}{12} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{175}{24} olish uchun. Keyin, -\frac{175}{24} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{175}{12}x+\left(-\frac{175}{24}\right)^{2}=-\frac{575}{9}+\left(-\frac{175}{24}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{175}{24} kvadratini chiqarish.

$$x^{2}-\frac{175}{12}x+\frac{30625}{576}=-\frac{575}{9}+\frac{30625}{576}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{575}{9} ni \frac{30625}{576} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{175}{12}x+\frac{30625}{576}=-\frac{6175}{576}$$

x^{2}-\frac{175}{12}x+\frac{30625}{576} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{175}{24}\right)^{2}=-\frac{6175}{576}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{175}{24}\right)^{2}}=\sqrt{-\frac{6175}{576}}$$

Qisqartirish.

$$x-\frac{175}{24}=\frac{5\sqrt{247}i}{24} x-\frac{175}{24}=-\frac{5\sqrt{247}i}{24}$$

\frac{175}{24} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{175+5\sqrt{247}i}{24} x=\frac{-5\sqrt{247}i+175}{24}$$