14x^2+2x=3 eching

$$14 x ^ { 2 } + 2 x = 3$$

x uchun yechish

$$x=\frac{\sqrt{43}-1}{14}\approx 0,396959895$$

$$x=\frac{-\sqrt{43}-1}{14}\approx -0,539817037$$

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.

$$14x^{2}+2x=3$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$14x^{2}+2x-3=3-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$14x^{2}+2x-3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 14 ni a, 2 ni b va -3 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\times 14\left(-3\right)}}{2\times 14}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\times 14\left(-3\right)}}{2\times 14}$$

-4 ni 14 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-56\left(-3\right)}}{2\times 14}$$

-56 ni -3 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+168}}{2\times 14}$$

4 ni 168 ga qo'shish.

$$x=\frac{-2±\sqrt{172}}{2\times 14}$$

172 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{43}}{2\times 14}$$

2 ni 14 marotabaga ko'paytirish.

$$x=\frac{-2±2\sqrt{43}}{28}$$

x=\frac{-2±2\sqrt{43}}{28} tenglamasini yeching, bunda ± musbat. -2 ni 2\sqrt{43} ga qo'shish.

$$x=\frac{2\sqrt{43}-2}{28}$$

-2+2\sqrt{43} ni 28 ga bo'lish.

$$x=\frac{\sqrt{43}-1}{14}$$

x=\frac{-2±2\sqrt{43}}{28} tenglamasini yeching, bunda ± manfiy. -2 dan 2\sqrt{43} ni ayirish.

$$x=\frac{-2\sqrt{43}-2}{28}$$

-2-2\sqrt{43} ni 28 ga bo'lish.

$$x=\frac{-\sqrt{43}-1}{14}$$

Tenglama yechildi.

$$x=\frac{\sqrt{43}-1}{14} x=\frac{-\sqrt{43}-1}{14}$$

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.

$$14x^{2}+2x=3$$

Ikki tarafini 14 ga bo‘ling.

$$\frac{14x^{2}+2x}{14}=\frac{3}{14}$$

14 ga bo'lish 14 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{2}{14}x=\frac{3}{14}$$

\frac{2}{14} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{1}{7}x=\frac{3}{14}$$

\frac{1}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{14} olish uchun. Keyin, \frac{1}{14} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{1}{7}x+\left(\frac{1}{14}\right)^{2}=\frac{3}{14}+\left(\frac{1}{14}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{14} kvadratini chiqarish.

$$x^{2}+\frac{1}{7}x+\frac{1}{196}=\frac{3}{14}+\frac{1}{196}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{3}{14} ni \frac{1}{196} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{1}{7}x+\frac{1}{196}=\frac{43}{196}$$

x^{2}+\frac{1}{7}x+\frac{1}{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.

$$\left(x+\frac{1}{14}\right)^{2}=\frac{43}{196}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{14}\right)^{2}}=\sqrt{\frac{43}{196}}$$

Qisqartirish.

$$x+\frac{1}{14}=\frac{\sqrt{43}}{14} x+\frac{1}{14}=-\frac{\sqrt{43}}{14}$$

Tenglamaning ikkala tarafidan \frac{1}{14} ni ayirish.

$$x=\frac{\sqrt{43}-1}{14} x=\frac{-\sqrt{43}-1}{14}$$