(76-4x)(8+2x)=1080 eching

$$( 76 - 4 x ) ( 8 + 2 x ) = 1080$$

x uchun yechish (complex solution)

$$x=\frac{15+\sqrt{11}i}{2}\approx 7,5+1,658312395i$$

$$x=\frac{-\sqrt{11}i+15}{2}\approx 7,5-1,658312395i$$

Kvadrat tenglamalaridan foydalanish qoidalari

76-4x ga 8+2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$608+120x-8x^{2}=1080$$

Ikkala tarafdan 1080 ni ayirish.

$$608+120x-8x^{2}-1080=0$$

-472 olish uchun 608 dan 1080 ni ayirish.

$$-472+120x-8x^{2}=0$$

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.

$$-8x^{2}+120x-472=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -8 ni a, 120 ni b va -472 ni c bilan almashtiring.

$$x=\frac{-120±\sqrt{120^{2}-4\left(-8\right)\left(-472\right)}}{2\left(-8\right)}$$

120 kvadratini chiqarish.

$$x=\frac{-120±\sqrt{14400-4\left(-8\right)\left(-472\right)}}{2\left(-8\right)}$$

-4 ni -8 marotabaga ko'paytirish.

$$x=\frac{-120±\sqrt{14400+32\left(-472\right)}}{2\left(-8\right)}$$

32 ni -472 marotabaga ko'paytirish.

$$x=\frac{-120±\sqrt{14400-15104}}{2\left(-8\right)}$$

14400 ni -15104 ga qo'shish.

$$x=\frac{-120±\sqrt{-704}}{2\left(-8\right)}$$

-704 ning kvadrat ildizini chiqarish.

$$x=\frac{-120±8\sqrt{11}i}{2\left(-8\right)}$$

2 ni -8 marotabaga ko'paytirish.

$$x=\frac{-120±8\sqrt{11}i}{-16}$$

x=\frac{-120±8\sqrt{11}i}{-16} tenglamasini yeching, bunda ± musbat. -120 ni 8i\sqrt{11} ga qo'shish.

$$x=\frac{-120+8\sqrt{11}i}{-16}$$

-120+8i\sqrt{11} ni -16 ga bo'lish.

$$x=\frac{-\sqrt{11}i+15}{2}$$

x=\frac{-120±8\sqrt{11}i}{-16} tenglamasini yeching, bunda ± manfiy. -120 dan 8i\sqrt{11} ni ayirish.

$$x=\frac{-8\sqrt{11}i-120}{-16}$$

-120-8i\sqrt{11} ni -16 ga bo'lish.

$$x=\frac{15+\sqrt{11}i}{2}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{11}i+15}{2} x=\frac{15+\sqrt{11}i}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

76-4x ga 8+2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$608+120x-8x^{2}=1080$$

Ikkala tarafdan 608 ni ayirish.

$$120x-8x^{2}=1080-608$$

472 olish uchun 1080 dan 608 ni ayirish.

$$120x-8x^{2}=472$$

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

$$-8x^{2}+120x=472$$

Ikki tarafini -8 ga bo‘ling.

$$\frac{-8x^{2}+120x}{-8}=\frac{472}{-8}$$

-8 ga bo'lish -8 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{120}{-8}x=\frac{472}{-8}$$

120 ni -8 ga bo'lish.

$$x^{2}-15x=\frac{472}{-8}$$

472 ni -8 ga bo'lish.

$$x^{2}-15x=-59$$

-15 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{15}{2} olish uchun. Keyin, -\frac{15}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-59+\left(-\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{2} kvadratini chiqarish.

$$x^{2}-15x+\frac{225}{4}=-59+\frac{225}{4}$$

-59 ni \frac{225}{4} ga qo'shish.

$$x^{2}-15x+\frac{225}{4}=-\frac{11}{4}$$

x^{2}-15x+\frac{225}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{15}{2}\right)^{2}=-\frac{11}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{-\frac{11}{4}}$$

Qisqartirish.

$$x-\frac{15}{2}=\frac{\sqrt{11}i}{2} x-\frac{15}{2}=-\frac{\sqrt{11}i}{2}$$

\frac{15}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{15+\sqrt{11}i}{2} x=\frac{-\sqrt{11}i+15}{2}$$