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

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

x uchun yechish (complex solution)

$$x=\frac{2+\sqrt{46}i}{5}\approx 0,4+1,356465997i$$

$$x=\frac{-\sqrt{46}i+2}{5}\approx 0,4-1,356465997i$$

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.

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

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

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

-4 kvadratini chiqarish.

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

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-20\times 10}}{2\times 5}$$

-20 ni 10 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-200}}{2\times 5}$$

16 ni -200 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{-184}}{2\times 5}$$

-184 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±2\sqrt{46}i}{2\times 5}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±2\sqrt{46}i}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{4±2\sqrt{46}i}{10}$$

x=\frac{4±2\sqrt{46}i}{10} tenglamasini yeching, bunda ± musbat. 4 ni 2i\sqrt{46} ga qo'shish.

$$x=\frac{4+2\sqrt{46}i}{10}$$

4+2i\sqrt{46} ni 10 ga bo'lish.

$$x=\frac{2+\sqrt{46}i}{5}$$

x=\frac{4±2\sqrt{46}i}{10} tenglamasini yeching, bunda ± manfiy. 4 dan 2i\sqrt{46} ni ayirish.

$$x=\frac{-2\sqrt{46}i+4}{10}$$

4-2i\sqrt{46} ni 10 ga bo'lish.

$$x=\frac{-\sqrt{46}i+2}{5}$$

Tenglama yechildi.

$$x=\frac{2+\sqrt{46}i}{5} x=\frac{-\sqrt{46}i+2}{5}$$

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.

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

Tenglamaning ikkala tarafidan 10 ni ayirish.

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

O‘zidan 10 ayirilsa 0 qoladi.

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

Ikki tarafini 5 ga bo‘ling.

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

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

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

-10 ni 5 ga bo'lish.

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

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{2}{5} kvadratini chiqarish.

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

-2 ni \frac{4}{25} ga qo'shish.

$$x^{2}-\frac{4}{5}x+\frac{4}{25}=-\frac{46}{25}$$

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

$$\left(x-\frac{2}{5}\right)^{2}=-\frac{46}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{2}{5}\right)^{2}}=\sqrt{-\frac{46}{25}}$$

Qisqartirish.

$$x-\frac{2}{5}=\frac{\sqrt{46}i}{5} x-\frac{2}{5}=-\frac{\sqrt{46}i}{5}$$

\frac{2}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2+\sqrt{46}i}{5} x=\frac{-\sqrt{46}i+2}{5}$$