3x^2*16-24x+24=0 eching

$$3 { x }^{ 2 } \times 16-24x+24=0$$

x uchun yechish (complex solution)

$$x=\frac{1+\sqrt{7}i}{4}\approx 0,25+0,661437828i$$

$$x=\frac{-\sqrt{7}i+1}{4}\approx 0,25-0,661437828i$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$48x^{2}-24x+24=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 48 ni a, -24 ni b va 24 ni c bilan almashtiring.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 48\times 24}}{2\times 48}$$

-24 kvadratini chiqarish.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 48\times 24}}{2\times 48}$$

-4 ni 48 marotabaga ko'paytirish.

$$x=\frac{-\left(-24\right)±\sqrt{576-192\times 24}}{2\times 48}$$

-192 ni 24 marotabaga ko'paytirish.

$$x=\frac{-\left(-24\right)±\sqrt{576-4608}}{2\times 48}$$

576 ni -4608 ga qo'shish.

$$x=\frac{-\left(-24\right)±\sqrt{-4032}}{2\times 48}$$

-4032 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-24\right)±24\sqrt{7}i}{2\times 48}$$

-24 ning teskarisi 24 ga teng.

$$x=\frac{24±24\sqrt{7}i}{2\times 48}$$

2 ni 48 marotabaga ko'paytirish.

$$x=\frac{24±24\sqrt{7}i}{96}$$

x=\frac{24±24\sqrt{7}i}{96} tenglamasini yeching, bunda ± musbat. 24 ni 24i\sqrt{7} ga qo'shish.

$$x=\frac{24+24\sqrt{7}i}{96}$$

24+24i\sqrt{7} ni 96 ga bo'lish.

$$x=\frac{1+\sqrt{7}i}{4}$$

x=\frac{24±24\sqrt{7}i}{96} tenglamasini yeching, bunda ± manfiy. 24 dan 24i\sqrt{7} ni ayirish.

$$x=\frac{-24\sqrt{7}i+24}{96}$$

24-24i\sqrt{7} ni 96 ga bo'lish.

$$x=\frac{-\sqrt{7}i+1}{4}$$

Tenglama yechildi.

$$x=\frac{1+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i+1}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$48x^{2}-24x+24=0$$

Tenglamaning ikkala tarafidan 24 ni ayirish.

$$48x^{2}-24x+24-24=-24$$

O‘zidan 24 ayirilsa 0 qoladi.

$$48x^{2}-24x=-24$$

Ikki tarafini 48 ga bo‘ling.

$$\frac{48x^{2}-24x}{48}=-\frac{24}{48}$$

48 ga bo'lish 48 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{24}{48}\right)x=-\frac{24}{48}$$

\frac{-24}{48} ulushini 24 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{2}x=-\frac{24}{48}$$

\frac{-24}{48} ulushini 24 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{2}x=-\frac{1}{2}$$

-\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{4} olish uchun. Keyin, -\frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{4} kvadratini chiqarish.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{1}{2}+\frac{1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{2} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{7}{16}$$

x^{2}-\frac{1}{2}x+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{7}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{7}{16}}$$

Qisqartirish.

$$x-\frac{1}{4}=\frac{\sqrt{7}i}{4} x-\frac{1}{4}=-\frac{\sqrt{7}i}{4}$$

\frac{1}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i+1}{4}$$