6x^2-15x+12=0 eching

$$6 x ^ { 2 } - 15 x + 12 = 0$$

x uchun yechish (complex solution)

$$x=\frac{5+\sqrt{7}i}{4}\approx 1,25+0,661437828i$$

$$x=\frac{-\sqrt{7}i+5}{4}\approx 1,25-0,661437828i$$

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.

$$6x^{2}-15x+12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, -15 ni b va 12 ni c bilan almashtiring.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 6\times 12}}{2\times 6}$$

-15 kvadratini chiqarish.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 6\times 12}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\left(-15\right)±\sqrt{225-24\times 12}}{2\times 6}$$

-24 ni 12 marotabaga ko'paytirish.

$$x=\frac{-\left(-15\right)±\sqrt{225-288}}{2\times 6}$$

225 ni -288 ga qo'shish.

$$x=\frac{-\left(-15\right)±\sqrt{-63}}{2\times 6}$$

-63 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-15\right)±3\sqrt{7}i}{2\times 6}$$

-15 ning teskarisi 15 ga teng.

$$x=\frac{15±3\sqrt{7}i}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$x=\frac{15±3\sqrt{7}i}{12}$$

x=\frac{15±3\sqrt{7}i}{12} tenglamasini yeching, bunda ± musbat. 15 ni 3i\sqrt{7} ga qo'shish.

$$x=\frac{15+3\sqrt{7}i}{12}$$

15+3i\sqrt{7} ni 12 ga bo'lish.

$$x=\frac{5+\sqrt{7}i}{4}$$

x=\frac{15±3\sqrt{7}i}{12} tenglamasini yeching, bunda ± manfiy. 15 dan 3i\sqrt{7} ni ayirish.

$$x=\frac{-3\sqrt{7}i+15}{12}$$

15-3i\sqrt{7} ni 12 ga bo'lish.

$$x=\frac{-\sqrt{7}i+5}{4}$$

Tenglama yechildi.

$$x=\frac{5+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i+5}{4}$$

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.

$$6x^{2}-15x+12=0$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$6x^{2}-15x+12-12=-12$$

O‘zidan 12 ayirilsa 0 qoladi.

$$6x^{2}-15x=-12$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6x^{2}-15x}{6}=-\frac{12}{6}$$

6 ga bo'lish 6 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{15}{6}\right)x=-\frac{12}{6}$$

\frac{-15}{6} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{5}{2}x=-\frac{12}{6}$$

-12 ni 6 ga bo'lish.

$$x^{2}-\frac{5}{2}x=-2$$

-\frac{5}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{4} olish uchun. Keyin, -\frac{5}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{4} kvadratini chiqarish.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=-2+\frac{25}{16}$$

-2 ni \frac{25}{16} ga qo'shish.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=-\frac{7}{16}$$

x^{2}-\frac{5}{2}x+\frac{25}{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.

$$\left(x-\frac{5}{4}\right)^{2}=-\frac{7}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{-\frac{7}{16}}$$

Qisqartirish.

$$x-\frac{5}{4}=\frac{\sqrt{7}i}{4} x-\frac{5}{4}=-\frac{\sqrt{7}i}{4}$$

\frac{5}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i+5}{4}$$