5x^2+7x=-3= eching

$$5 { x }^{ 2 } +7x = -3 =$$

x uchun yechish (complex solution)

$$x=\frac{-7+\sqrt{11}i}{10}\approx -0,7+0,331662479i$$

$$x=\frac{-\sqrt{11}i-7}{10}\approx -0,7-0,331662479i$$

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.

$$5x^{2}+7x=-3$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$5x^{2}+7x-\left(-3\right)=-3-\left(-3\right)$$

O‘zidan -3 ayirilsa 0 qoladi.

$$5x^{2}+7x-\left(-3\right)=0$$

0 dan -3 ni ayirish.

$$5x^{2}+7x+3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, 7 ni b va 3 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\times 5\times 3}}{2\times 5}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\times 5\times 3}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-20\times 3}}{2\times 5}$$

-20 ni 3 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-60}}{2\times 5}$$

49 ni -60 ga qo'shish.

$$x=\frac{-7±\sqrt{-11}}{2\times 5}$$

-11 ning kvadrat ildizini chiqarish.

$$x=\frac{-7±\sqrt{11}i}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{11}i}{10}$$

x=\frac{-7±\sqrt{11}i}{10} tenglamasini yeching, bunda ± musbat. -7 ni i\sqrt{11} ga qo'shish.

$$x=\frac{-7+\sqrt{11}i}{10}$$

x=\frac{-7±\sqrt{11}i}{10} tenglamasini yeching, bunda ± manfiy. -7 dan i\sqrt{11} ni ayirish.

$$x=\frac{-\sqrt{11}i-7}{10}$$

Tenglama yechildi.

$$x=\frac{-7+\sqrt{11}i}{10} x=\frac{-\sqrt{11}i-7}{10}$$

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.

$$5x^{2}+7x=-3$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}+7x}{5}=-\frac{3}{5}$$

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

$$x^{2}+\frac{7}{5}x=-\frac{3}{5}$$

\frac{7}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{10} olish uchun. Keyin, \frac{7}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{7}{5}x+\left(\frac{7}{10}\right)^{2}=-\frac{3}{5}+\left(\frac{7}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{10} kvadratini chiqarish.

$$x^{2}+\frac{7}{5}x+\frac{49}{100}=-\frac{3}{5}+\frac{49}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{5} ni \frac{49}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{7}{5}x+\frac{49}{100}=-\frac{11}{100}$$

x^{2}+\frac{7}{5}x+\frac{49}{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.

$$\left(x+\frac{7}{10}\right)^{2}=-\frac{11}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{10}\right)^{2}}=\sqrt{-\frac{11}{100}}$$

Qisqartirish.

$$x+\frac{7}{10}=\frac{\sqrt{11}i}{10} x+\frac{7}{10}=-\frac{\sqrt{11}i}{10}$$

Tenglamaning ikkala tarafidan \frac{7}{10} ni ayirish.

$$x=\frac{-7+\sqrt{11}i}{10} x=\frac{-\sqrt{11}i-7}{10}$$