(x+5)(x-8)=2x(x+5)+3x(x-8) eching

$$( x + 5 ) ( x - 8 ) = 2 x ( x + 5 ) + 3 x ( x - 8 )$$

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{519}i+11}{8}\approx 1,375-2,847696437i$$

$$x=\frac{11+\sqrt{519}i}{8}\approx 1,375+2,847696437i$$

Kvadrat tenglamalaridan foydalanish qoidalari

x+5 ga x-8 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}-3x-40=2x\left(x+5\right)+3x\left(x-8\right)$$

2x ga x+5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-3x-40=2x^{2}+10x+3x\left(x-8\right)$$

3x ga x-8 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-3x-40=2x^{2}+10x+3x^{2}-24x$$

5x^{2} ni olish uchun 2x^{2} va 3x^{2} ni birlashtirish.

$$x^{2}-3x-40=5x^{2}+10x-24x$$

-14x ni olish uchun 10x va -24x ni birlashtirish.

$$x^{2}-3x-40=5x^{2}-14x$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$x^{2}-3x-40-5x^{2}=-14x$$

-4x^{2} ni olish uchun x^{2} va -5x^{2} ni birlashtirish.

$$-4x^{2}-3x-40=-14x$$

14x ni ikki tarafga qo’shing.

$$-4x^{2}-3x-40+14x=0$$

11x ni olish uchun -3x va 14x ni birlashtirish.

$$-4x^{2}+11x-40=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -4 ni a, 11 ni b va -40 ni c bilan almashtiring.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-4\right)\left(-40\right)}}{2\left(-4\right)}$$

11 kvadratini chiqarish.

$$x=\frac{-11±\sqrt{121-4\left(-4\right)\left(-40\right)}}{2\left(-4\right)}$$

-4 ni -4 marotabaga ko'paytirish.

$$x=\frac{-11±\sqrt{121+16\left(-40\right)}}{2\left(-4\right)}$$

16 ni -40 marotabaga ko'paytirish.

$$x=\frac{-11±\sqrt{121-640}}{2\left(-4\right)}$$

121 ni -640 ga qo'shish.

$$x=\frac{-11±\sqrt{-519}}{2\left(-4\right)}$$

-519 ning kvadrat ildizini chiqarish.

$$x=\frac{-11±\sqrt{519}i}{2\left(-4\right)}$$

2 ni -4 marotabaga ko'paytirish.

$$x=\frac{-11±\sqrt{519}i}{-8}$$

x=\frac{-11±\sqrt{519}i}{-8} tenglamasini yeching, bunda ± musbat. -11 ni i\sqrt{519} ga qo'shish.

$$x=\frac{-11+\sqrt{519}i}{-8}$$

-11+i\sqrt{519} ni -8 ga bo'lish.

$$x=\frac{-\sqrt{519}i+11}{8}$$

x=\frac{-11±\sqrt{519}i}{-8} tenglamasini yeching, bunda ± manfiy. -11 dan i\sqrt{519} ni ayirish.

$$x=\frac{-\sqrt{519}i-11}{-8}$$

-11-i\sqrt{519} ni -8 ga bo'lish.

$$x=\frac{11+\sqrt{519}i}{8}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{519}i+11}{8} x=\frac{11+\sqrt{519}i}{8}$$

Kvadratni yakunlash bo'yicha qadamlar

x+5 ga x-8 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$x^{2}-3x-40=2x\left(x+5\right)+3x\left(x-8\right)$$

2x ga x+5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-3x-40=2x^{2}+10x+3x\left(x-8\right)$$

3x ga x-8 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x^{2}-3x-40=2x^{2}+10x+3x^{2}-24x$$

5x^{2} ni olish uchun 2x^{2} va 3x^{2} ni birlashtirish.

$$x^{2}-3x-40=5x^{2}+10x-24x$$

-14x ni olish uchun 10x va -24x ni birlashtirish.

$$x^{2}-3x-40=5x^{2}-14x$$

Ikkala tarafdan 5x^{2} ni ayirish.

$$x^{2}-3x-40-5x^{2}=-14x$$

-4x^{2} ni olish uchun x^{2} va -5x^{2} ni birlashtirish.

$$-4x^{2}-3x-40=-14x$$

14x ni ikki tarafga qo’shing.

$$-4x^{2}-3x-40+14x=0$$

11x ni olish uchun -3x va 14x ni birlashtirish.

$$-4x^{2}+11x-40=0$$

40 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$-4x^{2}+11x=40$$

Ikki tarafini -4 ga bo‘ling.

$$\frac{-4x^{2}+11x}{-4}=\frac{40}{-4}$$

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

$$x^{2}+\frac{11}{-4}x=\frac{40}{-4}$$

11 ni -4 ga bo'lish.

$$x^{2}-\frac{11}{4}x=\frac{40}{-4}$$

40 ni -4 ga bo'lish.

$$x^{2}-\frac{11}{4}x=-10$$

-\frac{11}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{8} olish uchun. Keyin, -\frac{11}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{11}{4}x+\left(-\frac{11}{8}\right)^{2}=-10+\left(-\frac{11}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{8} kvadratini chiqarish.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=-10+\frac{121}{64}$$

-10 ni \frac{121}{64} ga qo'shish.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=-\frac{519}{64}$$

x^{2}-\frac{11}{4}x+\frac{121}{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.

$$\left(x-\frac{11}{8}\right)^{2}=-\frac{519}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{8}\right)^{2}}=\sqrt{-\frac{519}{64}}$$

Qisqartirish.

$$x-\frac{11}{8}=\frac{\sqrt{519}i}{8} x-\frac{11}{8}=-\frac{\sqrt{519}i}{8}$$

\frac{11}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{11+\sqrt{519}i}{8} x=\frac{-\sqrt{519}i+11}{8}$$