12x^2-88x+400=0 eching

$$12 { x }^{ 2 } -88x+400 = 0$$

x uchun yechish (complex solution)

$$x=\frac{11+\sqrt{179}i}{3}\approx 3,666666667+4,459696053i$$

$$x=\frac{-\sqrt{179}i+11}{3}\approx 3,666666667-4,459696053i$$

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.

$$12x^{2}-88x+400=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 12 ni a, -88 ni b va 400 ni c bilan almashtiring.

$$x=\frac{-\left(-88\right)±\sqrt{\left(-88\right)^{2}-4\times 12\times 400}}{2\times 12}$$

-88 kvadratini chiqarish.

$$x=\frac{-\left(-88\right)±\sqrt{7744-4\times 12\times 400}}{2\times 12}$$

-4 ni 12 marotabaga ko'paytirish.

$$x=\frac{-\left(-88\right)±\sqrt{7744-48\times 400}}{2\times 12}$$

-48 ni 400 marotabaga ko'paytirish.

$$x=\frac{-\left(-88\right)±\sqrt{7744-19200}}{2\times 12}$$

7744 ni -19200 ga qo'shish.

$$x=\frac{-\left(-88\right)±\sqrt{-11456}}{2\times 12}$$

-11456 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-88\right)±8\sqrt{179}i}{2\times 12}$$

-88 ning teskarisi 88 ga teng.

$$x=\frac{88±8\sqrt{179}i}{2\times 12}$$

2 ni 12 marotabaga ko'paytirish.

$$x=\frac{88±8\sqrt{179}i}{24}$$

x=\frac{88±8\sqrt{179}i}{24} tenglamasini yeching, bunda ± musbat. 88 ni 8i\sqrt{179} ga qo'shish.

$$x=\frac{88+8\sqrt{179}i}{24}$$

88+8i\sqrt{179} ni 24 ga bo'lish.

$$x=\frac{11+\sqrt{179}i}{3}$$

x=\frac{88±8\sqrt{179}i}{24} tenglamasini yeching, bunda ± manfiy. 88 dan 8i\sqrt{179} ni ayirish.

$$x=\frac{-8\sqrt{179}i+88}{24}$$

88-8i\sqrt{179} ni 24 ga bo'lish.

$$x=\frac{-\sqrt{179}i+11}{3}$$

Tenglama yechildi.

$$x=\frac{11+\sqrt{179}i}{3} x=\frac{-\sqrt{179}i+11}{3}$$

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.

$$12x^{2}-88x+400=0$$

Tenglamaning ikkala tarafidan 400 ni ayirish.

$$12x^{2}-88x+400-400=-400$$

O‘zidan 400 ayirilsa 0 qoladi.

$$12x^{2}-88x=-400$$

Ikki tarafini 12 ga bo‘ling.

$$\frac{12x^{2}-88x}{12}=-\frac{400}{12}$$

12 ga bo'lish 12 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{88}{12}\right)x=-\frac{400}{12}$$

\frac{-88}{12} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{22}{3}x=-\frac{400}{12}$$

\frac{-400}{12} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{22}{3}x=-\frac{100}{3}$$

-\frac{22}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{3} olish uchun. Keyin, -\frac{11}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{22}{3}x+\left(-\frac{11}{3}\right)^{2}=-\frac{100}{3}+\left(-\frac{11}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{3} kvadratini chiqarish.

$$x^{2}-\frac{22}{3}x+\frac{121}{9}=-\frac{100}{3}+\frac{121}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{100}{3} ni \frac{121}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{22}{3}x+\frac{121}{9}=-\frac{179}{9}$$

x^{2}-\frac{22}{3}x+\frac{121}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{11}{3}\right)^{2}=-\frac{179}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{3}\right)^{2}}=\sqrt{-\frac{179}{9}}$$

Qisqartirish.

$$x-\frac{11}{3}=\frac{\sqrt{179}i}{3} x-\frac{11}{3}=-\frac{\sqrt{179}i}{3}$$

\frac{11}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{11+\sqrt{179}i}{3} x=\frac{-\sqrt{179}i+11}{3}$$