5x^2-2x+15=0 eching

$$5 { x }^{ 2 } -2x+15=0$$

x uchun yechish (complex solution)

$$x=\frac{1+\sqrt{74}i}{5}\approx 0,2+1,720465053i$$

$$x=\frac{-\sqrt{74}i+1}{5}\approx 0,2-1,720465053i$$

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.

$$5x^{2}-2x+15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, -2 ni b va 15 ni c bilan almashtiring.

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

-2 kvadratini chiqarish.

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

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-20\times 15}}{2\times 5}$$

-20 ni 15 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-300}}{2\times 5}$$

4 ni -300 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{-296}}{2\times 5}$$

-296 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±2\sqrt{74}i}{2\times 5}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±2\sqrt{74}i}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{2±2\sqrt{74}i}{10}$$

x=\frac{2±2\sqrt{74}i}{10} tenglamasini yeching, bunda ± musbat. 2 ni 2i\sqrt{74} ga qo'shish.

$$x=\frac{2+2\sqrt{74}i}{10}$$

2+2i\sqrt{74} ni 10 ga bo'lish.

$$x=\frac{1+\sqrt{74}i}{5}$$

x=\frac{2±2\sqrt{74}i}{10} tenglamasini yeching, bunda ± manfiy. 2 dan 2i\sqrt{74} ni ayirish.

$$x=\frac{-2\sqrt{74}i+2}{10}$$

2-2i\sqrt{74} ni 10 ga bo'lish.

$$x=\frac{-\sqrt{74}i+1}{5}$$

Tenglama yechildi.

$$x=\frac{1+\sqrt{74}i}{5} x=\frac{-\sqrt{74}i+1}{5}$$

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.

$$5x^{2}-2x+15=0$$

Tenglamaning ikkala tarafidan 15 ni ayirish.

$$5x^{2}-2x+15-15=-15$$

O‘zidan 15 ayirilsa 0 qoladi.

$$5x^{2}-2x=-15$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}-2x}{5}=-\frac{15}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{2}{5}x=-\frac{15}{5}$$

-15 ni 5 ga bo'lish.

$$x^{2}-\frac{2}{5}x=-3$$

-\frac{2}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{5} olish uchun. Keyin, -\frac{1}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=-3+\left(-\frac{1}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{5} kvadratini chiqarish.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=-3+\frac{1}{25}$$

-3 ni \frac{1}{25} ga qo'shish.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=-\frac{74}{25}$$

x^{2}-\frac{2}{5}x+\frac{1}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{5}\right)^{2}=-\frac{74}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{-\frac{74}{25}}$$

Qisqartirish.

$$x-\frac{1}{5}=\frac{\sqrt{74}i}{5} x-\frac{1}{5}=-\frac{\sqrt{74}i}{5}$$

\frac{1}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{74}i}{5} x=\frac{-\sqrt{74}i+1}{5}$$