4x^2-5x+10=0 eching

$$4 { x }^{ 2 } -5x+10=0$$

x uchun yechish (complex solution)

$$x=\frac{5+3\sqrt{15}i}{8}\approx 0,625+1,452368755i$$

$$x=\frac{-3\sqrt{15}i+5}{8}\approx 0,625-1,452368755i$$

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.

$$4x^{2}-5x+10=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -5 ni b va 10 ni c bilan almashtiring.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 4\times 10}}{2\times 4}$$

-5 kvadratini chiqarish.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 4\times 10}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25-16\times 10}}{2\times 4}$$

-16 ni 10 marotabaga ko'paytirish.

$$x=\frac{-\left(-5\right)±\sqrt{25-160}}{2\times 4}$$

25 ni -160 ga qo'shish.

$$x=\frac{-\left(-5\right)±\sqrt{-135}}{2\times 4}$$

-135 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-5\right)±3\sqrt{15}i}{2\times 4}$$

-5 ning teskarisi 5 ga teng.

$$x=\frac{5±3\sqrt{15}i}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{5±3\sqrt{15}i}{8}$$

x=\frac{5±3\sqrt{15}i}{8} tenglamasini yeching, bunda ± musbat. 5 ni 3i\sqrt{15} ga qo'shish.

$$x=\frac{5+3\sqrt{15}i}{8}$$

x=\frac{5±3\sqrt{15}i}{8} tenglamasini yeching, bunda ± manfiy. 5 dan 3i\sqrt{15} ni ayirish.

$$x=\frac{-3\sqrt{15}i+5}{8}$$

Tenglama yechildi.

$$x=\frac{5+3\sqrt{15}i}{8} x=\frac{-3\sqrt{15}i+5}{8}$$

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.

$$4x^{2}-5x+10=0$$

Tenglamaning ikkala tarafidan 10 ni ayirish.

$$4x^{2}-5x+10-10=-10$$

O‘zidan 10 ayirilsa 0 qoladi.

$$4x^{2}-5x=-10$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-5x}{4}=-\frac{10}{4}$$

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

$$x^{2}-\frac{5}{4}x=-\frac{10}{4}$$

\frac{-10}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{5}{4}x=-\frac{5}{2}$$

-\frac{5}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{8} olish uchun. Keyin, -\frac{5}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=-\frac{5}{2}+\left(-\frac{5}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{8} kvadratini chiqarish.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-\frac{5}{2}+\frac{25}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{2} ni \frac{25}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-\frac{135}{64}$$

x^{2}-\frac{5}{4}x+\frac{25}{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.

$$\left(x-\frac{5}{8}\right)^{2}=-\frac{135}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{-\frac{135}{64}}$$

Qisqartirish.

$$x-\frac{5}{8}=\frac{3\sqrt{15}i}{8} x-\frac{5}{8}=-\frac{3\sqrt{15}i}{8}$$

\frac{5}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5+3\sqrt{15}i}{8} x=\frac{-3\sqrt{15}i+5}{8}$$