7x^2+5x+5=0 eching

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

x uchun yechish (complex solution)

$$x=\frac{-5+\sqrt{115}i}{14}\approx -0,357142857+0,765986092i$$

$$x=\frac{-\sqrt{115}i-5}{14}\approx -0,357142857-0,765986092i$$

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.

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

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

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

5 kvadratini chiqarish.

$$x=\frac{-5±\sqrt{25-4\times 7\times 5}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25-28\times 5}}{2\times 7}$$

-28 ni 5 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{25-140}}{2\times 7}$$

25 ni -140 ga qo'shish.

$$x=\frac{-5±\sqrt{-115}}{2\times 7}$$

-115 ning kvadrat ildizini chiqarish.

$$x=\frac{-5±\sqrt{115}i}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$x=\frac{-5±\sqrt{115}i}{14}$$

x=\frac{-5±\sqrt{115}i}{14} tenglamasini yeching, bunda ± musbat. -5 ni i\sqrt{115} ga qo'shish.

$$x=\frac{-5+\sqrt{115}i}{14}$$

x=\frac{-5±\sqrt{115}i}{14} tenglamasini yeching, bunda ± manfiy. -5 dan i\sqrt{115} ni ayirish.

$$x=\frac{-\sqrt{115}i-5}{14}$$

Tenglama yechildi.

$$x=\frac{-5+\sqrt{115}i}{14} x=\frac{-\sqrt{115}i-5}{14}$$

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.

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

Tenglamaning ikkala tarafidan 5 ni ayirish.

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

O‘zidan 5 ayirilsa 0 qoladi.

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

Ikki tarafini 7 ga bo‘ling.

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

7 ga bo'lish 7 ga ko'paytirishni bekor qiladi.

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

\frac{5}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{14} olish uchun. Keyin, \frac{5}{14} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{5}{7}x+\left(\frac{5}{14}\right)^{2}=-\frac{5}{7}+\left(\frac{5}{14}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{14} kvadratini chiqarish.

$$x^{2}+\frac{5}{7}x+\frac{25}{196}=-\frac{5}{7}+\frac{25}{196}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{7} ni \frac{25}{196} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{5}{7}x+\frac{25}{196}=-\frac{115}{196}$$

x^{2}+\frac{5}{7}x+\frac{25}{196} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{5}{14}\right)^{2}=-\frac{115}{196}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{14}\right)^{2}}=\sqrt{-\frac{115}{196}}$$

Qisqartirish.

$$x+\frac{5}{14}=\frac{\sqrt{115}i}{14} x+\frac{5}{14}=-\frac{\sqrt{115}i}{14}$$

Tenglamaning ikkala tarafidan \frac{5}{14} ni ayirish.

$$x=\frac{-5+\sqrt{115}i}{14} x=\frac{-\sqrt{115}i-5}{14}$$