4x^2-52x+71=0 eching

$$4 x ^ { 2 } - 52 x + 71 = 0$$

x uchun yechish

$$x = \frac{7 \sqrt{2} + 13}{2} \approx 11,449747468$$

$$x = \frac{13 - 7 \sqrt{2}}{2} \approx 1,550252532$$

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.

$$4x^{2}-52x+71=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -52 ni b va 71 ni c bilan almashtiring.

$$x=\frac{-\left(-52\right)±\sqrt{\left(-52\right)^{2}-4\times 4\times 71}}{2\times 4}$$

-52 kvadratini chiqarish.

$$x=\frac{-\left(-52\right)±\sqrt{2704-4\times 4\times 71}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-52\right)±\sqrt{2704-16\times 71}}{2\times 4}$$

-16 ni 71 marotabaga ko'paytirish.

$$x=\frac{-\left(-52\right)±\sqrt{2704-1136}}{2\times 4}$$

2704 ni -1136 ga qo'shish.

$$x=\frac{-\left(-52\right)±\sqrt{1568}}{2\times 4}$$

1568 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-52\right)±28\sqrt{2}}{2\times 4}$$

-52 ning teskarisi 52 ga teng.

$$x=\frac{52±28\sqrt{2}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{52±28\sqrt{2}}{8}$$

x=\frac{52±28\sqrt{2}}{8} tenglamasini yeching, bunda ± musbat. 52 ni 28\sqrt{2} ga qo'shish.

$$x=\frac{28\sqrt{2}+52}{8}$$

52+28\sqrt{2} ni 8 ga bo'lish.

$$x=\frac{7\sqrt{2}+13}{2}$$

x=\frac{52±28\sqrt{2}}{8} tenglamasini yeching, bunda ± manfiy. 52 dan 28\sqrt{2} ni ayirish.

$$x=\frac{52-28\sqrt{2}}{8}$$

52-28\sqrt{2} ni 8 ga bo'lish.

$$x=\frac{13-7\sqrt{2}}{2}$$

Tenglama yechildi.

$$x=\frac{7\sqrt{2}+13}{2} x=\frac{13-7\sqrt{2}}{2}$$

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.

$$4x^{2}-52x+71=0$$

Tenglamaning ikkala tarafidan 71 ni ayirish.

$$4x^{2}-52x+71-71=-71$$

O‘zidan 71 ayirilsa 0 qoladi.

$$4x^{2}-52x=-71$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-52x}{4}=-\frac{71}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{52}{4}\right)x=-\frac{71}{4}$$

-52 ni 4 ga bo'lish.

$$x^{2}-13x=-\frac{71}{4}$$

-13 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{13}{2} olish uchun. Keyin, -\frac{13}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-\frac{71}{4}+\left(-\frac{13}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{13}{2} kvadratini chiqarish.

$$x^{2}-13x+\frac{169}{4}=\frac{-71+169}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{71}{4} ni \frac{169}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-13x+\frac{169}{4}=\frac{49}{2}$$

x^{2}-13x+\frac{169}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{49}{2}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{49}{2}}$$

Qisqartirish.

$$x-\frac{13}{2}=\frac{7\sqrt{2}}{2} x-\frac{13}{2}=-\frac{7\sqrt{2}}{2}$$

\frac{13}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{7\sqrt{2}+13}{2} x=\frac{13-7\sqrt{2}}{2}$$