3x-5x^2=3 eching

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

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{51}i+3}{10}\approx 0,3-0,714142843i$$

$$x=\frac{3+\sqrt{51}i}{10}\approx 0,3+0,714142843i$$

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.

$$-5x^{2}+3x=3$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$-5x^{2}+3x-3=3-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$-5x^{2}+3x-3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -5 ni a, 3 ni b va -3 ni c bilan almashtiring.

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

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\left(-5\right)\left(-3\right)}}{2\left(-5\right)}$$

-4 ni -5 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+20\left(-3\right)}}{2\left(-5\right)}$$

20 ni -3 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9-60}}{2\left(-5\right)}$$

9 ni -60 ga qo'shish.

$$x=\frac{-3±\sqrt{-51}}{2\left(-5\right)}$$

-51 ning kvadrat ildizini chiqarish.

$$x=\frac{-3±\sqrt{51}i}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{51}i}{-10}$$

x=\frac{-3±\sqrt{51}i}{-10} tenglamasini yeching, bunda ± musbat. -3 ni i\sqrt{51} ga qo'shish.

$$x=\frac{-3+\sqrt{51}i}{-10}$$

-3+i\sqrt{51} ni -10 ga bo'lish.

$$x=\frac{-\sqrt{51}i+3}{10}$$

x=\frac{-3±\sqrt{51}i}{-10} tenglamasini yeching, bunda ± manfiy. -3 dan i\sqrt{51} ni ayirish.

$$x=\frac{-\sqrt{51}i-3}{-10}$$

-3-i\sqrt{51} ni -10 ga bo'lish.

$$x=\frac{3+\sqrt{51}i}{10}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{51}i+3}{10} x=\frac{3+\sqrt{51}i}{10}$$

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.

$$-5x^{2}+3x=3$$

Ikki tarafini -5 ga bo‘ling.

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

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

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

3 ni -5 ga bo'lish.

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

3 ni -5 ga bo'lish.

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

-\frac{3}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{10} olish uchun. Keyin, -\frac{3}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{5}x+\left(-\frac{3}{10}\right)^{2}=-\frac{3}{5}+\left(-\frac{3}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{10} kvadratini chiqarish.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=-\frac{3}{5}+\frac{9}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{5} ni \frac{9}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{5}x+\frac{9}{100}=-\frac{51}{100}$$

x^{2}-\frac{3}{5}x+\frac{9}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{10}\right)^{2}=-\frac{51}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{10}\right)^{2}}=\sqrt{-\frac{51}{100}}$$

Qisqartirish.

$$x-\frac{3}{10}=\frac{\sqrt{51}i}{10} x-\frac{3}{10}=-\frac{\sqrt{51}i}{10}$$

\frac{3}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3+\sqrt{51}i}{10} x=\frac{-\sqrt{51}i+3}{10}$$