2x-49x^2=3 eching

$$2 x - 49 x ^ { 2 } = 3$$

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{146}i+1}{49}\approx 0,020408163-0,246592775i$$

$$x=\frac{1+\sqrt{146}i}{49}\approx 0,020408163+0,246592775i$$

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.

$$-49x^{2}+2x=3$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$-49x^{2}+2x-3=3-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$-49x^{2}+2x-3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -49 ni a, 2 ni b va -3 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-49\right)\left(-3\right)}}{2\left(-49\right)}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\left(-49\right)\left(-3\right)}}{2\left(-49\right)}$$

-4 ni -49 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+196\left(-3\right)}}{2\left(-49\right)}$$

196 ni -3 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-588}}{2\left(-49\right)}$$

4 ni -588 ga qo'shish.

$$x=\frac{-2±\sqrt{-584}}{2\left(-49\right)}$$

-584 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{146}i}{2\left(-49\right)}$$

2 ni -49 marotabaga ko'paytirish.

$$x=\frac{-2±2\sqrt{146}i}{-98}$$

x=\frac{-2±2\sqrt{146}i}{-98} tenglamasini yeching, bunda ± musbat. -2 ni 2i\sqrt{146} ga qo'shish.

$$x=\frac{-2+2\sqrt{146}i}{-98}$$

-2+2i\sqrt{146} ni -98 ga bo'lish.

$$x=\frac{-\sqrt{146}i+1}{49}$$

x=\frac{-2±2\sqrt{146}i}{-98} tenglamasini yeching, bunda ± manfiy. -2 dan 2i\sqrt{146} ni ayirish.

$$x=\frac{-2\sqrt{146}i-2}{-98}$$

-2-2i\sqrt{146} ni -98 ga bo'lish.

$$x=\frac{1+\sqrt{146}i}{49}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{146}i+1}{49} x=\frac{1+\sqrt{146}i}{49}$$

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.

$$-49x^{2}+2x=3$$

Ikki tarafini -49 ga bo‘ling.

$$\frac{-49x^{2}+2x}{-49}=\frac{3}{-49}$$

-49 ga bo'lish -49 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{2}{-49}x=\frac{3}{-49}$$

2 ni -49 ga bo'lish.

$$x^{2}-\frac{2}{49}x=\frac{3}{-49}$$

3 ni -49 ga bo'lish.

$$x^{2}-\frac{2}{49}x=-\frac{3}{49}$$

-\frac{2}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{49} olish uchun. Keyin, -\frac{1}{49} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{2}{49}x+\left(-\frac{1}{49}\right)^{2}=-\frac{3}{49}+\left(-\frac{1}{49}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{49} kvadratini chiqarish.

$$x^{2}-\frac{2}{49}x+\frac{1}{2401}=-\frac{3}{49}+\frac{1}{2401}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{49} ni \frac{1}{2401} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{2}{49}x+\frac{1}{2401}=-\frac{146}{2401}$$

x^{2}-\frac{2}{49}x+\frac{1}{2401} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{49}\right)^{2}=-\frac{146}{2401}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{49}\right)^{2}}=\sqrt{-\frac{146}{2401}}$$

Qisqartirish.

$$x-\frac{1}{49}=\frac{\sqrt{146}i}{49} x-\frac{1}{49}=-\frac{\sqrt{146}i}{49}$$

\frac{1}{49} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{146}i}{49} x=\frac{-\sqrt{146}i+1}{49}$$