4x^2+13x+5=0 eching

$$4 { x }^{ 2 } +13x+5=0$$

x uchun yechish

$$x=\frac{\sqrt{89}-13}{8}\approx -0,445752358$$

$$x=\frac{-\sqrt{89}-13}{8}\approx -2,804247642$$

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.

$$4x^{2}+13x+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, 13 ni b va 5 ni c bilan almashtiring.

$$x=\frac{-13±\sqrt{13^{2}-4\times 4\times 5}}{2\times 4}$$

13 kvadratini chiqarish.

$$x=\frac{-13±\sqrt{169-4\times 4\times 5}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-13±\sqrt{169-16\times 5}}{2\times 4}$$

-16 ni 5 marotabaga ko'paytirish.

$$x=\frac{-13±\sqrt{169-80}}{2\times 4}$$

169 ni -80 ga qo'shish.

$$x=\frac{-13±\sqrt{89}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{-13±\sqrt{89}}{8}$$

x=\frac{-13±\sqrt{89}}{8} tenglamasini yeching, bunda ± musbat. -13 ni \sqrt{89} ga qo'shish.

$$x=\frac{\sqrt{89}-13}{8}$$

x=\frac{-13±\sqrt{89}}{8} tenglamasini yeching, bunda ± manfiy. -13 dan \sqrt{89} ni ayirish.

$$x=\frac{-\sqrt{89}-13}{8}$$

Tenglama yechildi.

$$x=\frac{\sqrt{89}-13}{8} x=\frac{-\sqrt{89}-13}{8}$$

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.

$$4x^{2}+13x+5=0$$

Tenglamaning ikkala tarafidan 5 ni ayirish.

$$4x^{2}+13x+5-5=-5$$

O‘zidan 5 ayirilsa 0 qoladi.

$$4x^{2}+13x=-5$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}+13x}{4}=-\frac{5}{4}$$

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

$$x^{2}+\frac{13}{4}x=-\frac{5}{4}$$

\frac{13}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{13}{8} olish uchun. Keyin, \frac{13}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{13}{4}x+\left(\frac{13}{8}\right)^{2}=-\frac{5}{4}+\left(\frac{13}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{13}{8} kvadratini chiqarish.

$$x^{2}+\frac{13}{4}x+\frac{169}{64}=-\frac{5}{4}+\frac{169}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{4} ni \frac{169}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{13}{4}x+\frac{169}{64}=\frac{89}{64}$$

x^{2}+\frac{13}{4}x+\frac{169}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{13}{8}\right)^{2}=\frac{89}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{13}{8}\right)^{2}}=\sqrt{\frac{89}{64}}$$

Qisqartirish.

$$x+\frac{13}{8}=\frac{\sqrt{89}}{8} x+\frac{13}{8}=-\frac{\sqrt{89}}{8}$$

Tenglamaning ikkala tarafidan \frac{13}{8} ni ayirish.

$$x=\frac{\sqrt{89}-13}{8} x=\frac{-\sqrt{89}-13}{8}$$