101x^2+7x+6=0 eching

$$101 x ^ { 2 } + 7 x + 6 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-7+5\sqrt{95}i}{202}\approx -0,034653465+0,241257286i$$

$$x=\frac{-5\sqrt{95}i-7}{202}\approx -0,034653465-0,241257286i$$

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.

$$101x^{2}+7x+6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 101 ni a, 7 ni b va 6 ni c bilan almashtiring.

$$x=\frac{-7±\sqrt{7^{2}-4\times 101\times 6}}{2\times 101}$$

7 kvadratini chiqarish.

$$x=\frac{-7±\sqrt{49-4\times 101\times 6}}{2\times 101}$$

-4 ni 101 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-404\times 6}}{2\times 101}$$

-404 ni 6 marotabaga ko'paytirish.

$$x=\frac{-7±\sqrt{49-2424}}{2\times 101}$$

49 ni -2424 ga qo'shish.

$$x=\frac{-7±\sqrt{-2375}}{2\times 101}$$

-2375 ning kvadrat ildizini chiqarish.

$$x=\frac{-7±5\sqrt{95}i}{2\times 101}$$

2 ni 101 marotabaga ko'paytirish.

$$x=\frac{-7±5\sqrt{95}i}{202}$$

x=\frac{-7±5\sqrt{95}i}{202} tenglamasini yeching, bunda ± musbat. -7 ni 5i\sqrt{95} ga qo'shish.

$$x=\frac{-7+5\sqrt{95}i}{202}$$

x=\frac{-7±5\sqrt{95}i}{202} tenglamasini yeching, bunda ± manfiy. -7 dan 5i\sqrt{95} ni ayirish.

$$x=\frac{-5\sqrt{95}i-7}{202}$$

Tenglama yechildi.

$$x=\frac{-7+5\sqrt{95}i}{202} x=\frac{-5\sqrt{95}i-7}{202}$$

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.

$$101x^{2}+7x+6=0$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$101x^{2}+7x+6-6=-6$$

O‘zidan 6 ayirilsa 0 qoladi.

$$101x^{2}+7x=-6$$

Ikki tarafini 101 ga bo‘ling.

$$\frac{101x^{2}+7x}{101}=-\frac{6}{101}$$

101 ga bo'lish 101 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{7}{101}x=-\frac{6}{101}$$

\frac{7}{101} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{202} olish uchun. Keyin, \frac{7}{202} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{7}{101}x+\left(\frac{7}{202}\right)^{2}=-\frac{6}{101}+\left(\frac{7}{202}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{202} kvadratini chiqarish.

$$x^{2}+\frac{7}{101}x+\frac{49}{40804}=-\frac{6}{101}+\frac{49}{40804}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{6}{101} ni \frac{49}{40804} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{7}{101}x+\frac{49}{40804}=-\frac{2375}{40804}$$

x^{2}+\frac{7}{101}x+\frac{49}{40804} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{7}{202}\right)^{2}=-\frac{2375}{40804}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{202}\right)^{2}}=\sqrt{-\frac{2375}{40804}}$$

Qisqartirish.

$$x+\frac{7}{202}=\frac{5\sqrt{95}i}{202} x+\frac{7}{202}=-\frac{5\sqrt{95}i}{202}$$

Tenglamaning ikkala tarafidan \frac{7}{202} ni ayirish.

$$x=\frac{-7+5\sqrt{95}i}{202} x=\frac{-5\sqrt{95}i-7}{202}$$