28x^2+13x+6=0 eching

$$28 x ^ { 2 } + 13 x + 6 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-13+\sqrt{503}i}{56}\approx -0,232142857+0,400493955i$$

$$x=\frac{-\sqrt{503}i-13}{56}\approx -0,232142857-0,400493955i$$

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.

$$28x^{2}+13x+6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 28 ni a, 13 ni b va 6 ni c bilan almashtiring.

$$x=\frac{-13±\sqrt{13^{2}-4\times 28\times 6}}{2\times 28}$$

13 kvadratini chiqarish.

$$x=\frac{-13±\sqrt{169-4\times 28\times 6}}{2\times 28}$$

-4 ni 28 marotabaga ko'paytirish.

$$x=\frac{-13±\sqrt{169-112\times 6}}{2\times 28}$$

-112 ni 6 marotabaga ko'paytirish.

$$x=\frac{-13±\sqrt{169-672}}{2\times 28}$$

169 ni -672 ga qo'shish.

$$x=\frac{-13±\sqrt{-503}}{2\times 28}$$

-503 ning kvadrat ildizini chiqarish.

$$x=\frac{-13±\sqrt{503}i}{2\times 28}$$

2 ni 28 marotabaga ko'paytirish.

$$x=\frac{-13±\sqrt{503}i}{56}$$

x=\frac{-13±\sqrt{503}i}{56} tenglamasini yeching, bunda ± musbat. -13 ni i\sqrt{503} ga qo'shish.

$$x=\frac{-13+\sqrt{503}i}{56}$$

x=\frac{-13±\sqrt{503}i}{56} tenglamasini yeching, bunda ± manfiy. -13 dan i\sqrt{503} ni ayirish.

$$x=\frac{-\sqrt{503}i-13}{56}$$

Tenglama yechildi.

$$x=\frac{-13+\sqrt{503}i}{56} x=\frac{-\sqrt{503}i-13}{56}$$

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.

$$28x^{2}+13x+6=0$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$28x^{2}+13x+6-6=-6$$

O‘zidan 6 ayirilsa 0 qoladi.

$$28x^{2}+13x=-6$$

Ikki tarafini 28 ga bo‘ling.

$$\frac{28x^{2}+13x}{28}=-\frac{6}{28}$$

28 ga bo'lish 28 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{13}{28}x=-\frac{6}{28}$$

\frac{-6}{28} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{13}{28}x=-\frac{3}{14}$$

\frac{13}{28} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{13}{56} olish uchun. Keyin, \frac{13}{56} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{13}{28}x+\left(\frac{13}{56}\right)^{2}=-\frac{3}{14}+\left(\frac{13}{56}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{13}{56} kvadratini chiqarish.

$$x^{2}+\frac{13}{28}x+\frac{169}{3136}=-\frac{3}{14}+\frac{169}{3136}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{14} ni \frac{169}{3136} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{13}{28}x+\frac{169}{3136}=-\frac{503}{3136}$$

x^{2}+\frac{13}{28}x+\frac{169}{3136} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{13}{56}\right)^{2}=-\frac{503}{3136}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{13}{56}\right)^{2}}=\sqrt{-\frac{503}{3136}}$$

Qisqartirish.

$$x+\frac{13}{56}=\frac{\sqrt{503}i}{56} x+\frac{13}{56}=-\frac{\sqrt{503}i}{56}$$

Tenglamaning ikkala tarafidan \frac{13}{56} ni ayirish.

$$x=\frac{-13+\sqrt{503}i}{56} x=\frac{-\sqrt{503}i-13}{56}$$