-25x^2+21x-5=0 eching

$$- 25 x ^ { 2 } + 21 x - 5 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{59}i+21}{50}\approx 0,42-0,153622915i$$

$$x=\frac{21+\sqrt{59}i}{50}\approx 0,42+0,153622915i$$

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.

$$-25x^{2}+21x-5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -25 ni a, 21 ni b va -5 ni c bilan almashtiring.

$$x=\frac{-21±\sqrt{21^{2}-4\left(-25\right)\left(-5\right)}}{2\left(-25\right)}$$

21 kvadratini chiqarish.

$$x=\frac{-21±\sqrt{441-4\left(-25\right)\left(-5\right)}}{2\left(-25\right)}$$

-4 ni -25 marotabaga ko'paytirish.

$$x=\frac{-21±\sqrt{441+100\left(-5\right)}}{2\left(-25\right)}$$

100 ni -5 marotabaga ko'paytirish.

$$x=\frac{-21±\sqrt{441-500}}{2\left(-25\right)}$$

441 ni -500 ga qo'shish.

$$x=\frac{-21±\sqrt{-59}}{2\left(-25\right)}$$

-59 ning kvadrat ildizini chiqarish.

$$x=\frac{-21±\sqrt{59}i}{2\left(-25\right)}$$

2 ni -25 marotabaga ko'paytirish.

$$x=\frac{-21±\sqrt{59}i}{-50}$$

x=\frac{-21±\sqrt{59}i}{-50} tenglamasini yeching, bunda ± musbat. -21 ni i\sqrt{59} ga qo'shish.

$$x=\frac{-21+\sqrt{59}i}{-50}$$

-21+i\sqrt{59} ni -50 ga bo'lish.

$$x=\frac{-\sqrt{59}i+21}{50}$$

x=\frac{-21±\sqrt{59}i}{-50} tenglamasini yeching, bunda ± manfiy. -21 dan i\sqrt{59} ni ayirish.

$$x=\frac{-\sqrt{59}i-21}{-50}$$

-21-i\sqrt{59} ni -50 ga bo'lish.

$$x=\frac{21+\sqrt{59}i}{50}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{59}i+21}{50} x=\frac{21+\sqrt{59}i}{50}$$

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.

$$-25x^{2}+21x-5=0$$

5 ni tenglamaning ikkala tarafiga qo'shish.

$$-25x^{2}+21x-5-\left(-5\right)=-\left(-5\right)$$

O‘zidan -5 ayirilsa 0 qoladi.

$$-25x^{2}+21x=-\left(-5\right)$$

0 dan -5 ni ayirish.

$$-25x^{2}+21x=5$$

Ikki tarafini -25 ga bo‘ling.

$$\frac{-25x^{2}+21x}{-25}=\frac{5}{-25}$$

-25 ga bo'lish -25 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{21}{-25}x=\frac{5}{-25}$$

21 ni -25 ga bo'lish.

$$x^{2}-\frac{21}{25}x=\frac{5}{-25}$$

\frac{5}{-25} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{21}{25}x=-\frac{1}{5}$$

-\frac{21}{25} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{21}{50} olish uchun. Keyin, -\frac{21}{50} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{21}{25}x+\left(-\frac{21}{50}\right)^{2}=-\frac{1}{5}+\left(-\frac{21}{50}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{21}{50} kvadratini chiqarish.

$$x^{2}-\frac{21}{25}x+\frac{441}{2500}=-\frac{1}{5}+\frac{441}{2500}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{5} ni \frac{441}{2500} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{21}{25}x+\frac{441}{2500}=-\frac{59}{2500}$$

x^{2}-\frac{21}{25}x+\frac{441}{2500} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{21}{50}\right)^{2}=-\frac{59}{2500}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{21}{50}\right)^{2}}=\sqrt{-\frac{59}{2500}}$$

Qisqartirish.

$$x-\frac{21}{50}=\frac{\sqrt{59}i}{50} x-\frac{21}{50}=-\frac{\sqrt{59}i}{50}$$

\frac{21}{50} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{21+\sqrt{59}i}{50} x=\frac{-\sqrt{59}i+21}{50}$$