x(x+3)/3=5x-1/4 eching

$$\frac { x ( x + 3 ) } { 3 } = \frac { 5 x - 1 } { 4 }$$

x uchun yechish (complex solution)

$$x=\frac{3+\sqrt{39}i}{8}\approx 0,375+0,78062475i$$

$$x=\frac{-\sqrt{39}i+3}{8}\approx 0,375-0,78062475i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 12 ga, 3,4 ning eng kichik karralisiga ko‘paytiring.

$$4x\left(x+3\right)=3\left(5x-1\right)$$

4x ga x+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}+12x=3\left(5x-1\right)$$

3 ga 5x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}+12x=15x-3$$

Ikkala tarafdan 15x ni ayirish.

$$4x^{2}+12x-15x=-3$$

-3x ni olish uchun 12x va -15x ni birlashtirish.

$$4x^{2}-3x=-3$$

3 ni ikki tarafga qo’shing.

$$4x^{2}-3x+3=0$$

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

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

-3 kvadratini chiqarish.

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

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-16\times 3}}{2\times 4}$$

-16 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-3\right)±\sqrt{9-48}}{2\times 4}$$

9 ni -48 ga qo'shish.

$$x=\frac{-\left(-3\right)±\sqrt{-39}}{2\times 4}$$

-39 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-3\right)±\sqrt{39}i}{2\times 4}$$

-3 ning teskarisi 3 ga teng.

$$x=\frac{3±\sqrt{39}i}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{3±\sqrt{39}i}{8}$$

x=\frac{3±\sqrt{39}i}{8} tenglamasini yeching, bunda ± musbat. 3 ni i\sqrt{39} ga qo'shish.

$$x=\frac{3+\sqrt{39}i}{8}$$

x=\frac{3±\sqrt{39}i}{8} tenglamasini yeching, bunda ± manfiy. 3 dan i\sqrt{39} ni ayirish.

$$x=\frac{-\sqrt{39}i+3}{8}$$

Tenglama yechildi.

$$x=\frac{3+\sqrt{39}i}{8} x=\frac{-\sqrt{39}i+3}{8}$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 12 ga, 3,4 ning eng kichik karralisiga ko‘paytiring.

$$4x\left(x+3\right)=3\left(5x-1\right)$$

4x ga x+3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}+12x=3\left(5x-1\right)$$

3 ga 5x-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}+12x=15x-3$$

Ikkala tarafdan 15x ni ayirish.

$$4x^{2}+12x-15x=-3$$

-3x ni olish uchun 12x va -15x ni birlashtirish.

$$4x^{2}-3x=-3$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-3x}{4}=-\frac{3}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{3}{4}x=-\frac{3}{4}$$

-\frac{3}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{8} olish uchun. Keyin, -\frac{3}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{4}x+\left(-\frac{3}{8}\right)^{2}=-\frac{3}{4}+\left(-\frac{3}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{8} kvadratini chiqarish.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=-\frac{3}{4}+\frac{9}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{4} ni \frac{9}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=-\frac{39}{64}$$

x^{2}-\frac{3}{4}x+\frac{9}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{8}\right)^{2}=-\frac{39}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{8}\right)^{2}}=\sqrt{-\frac{39}{64}}$$

Qisqartirish.

$$x-\frac{3}{8}=\frac{\sqrt{39}i}{8} x-\frac{3}{8}=-\frac{\sqrt{39}i}{8}$$

\frac{3}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3+\sqrt{39}i}{8} x=\frac{-\sqrt{39}i+3}{8}$$