225x^2-90x+324=0 eching

$$225 x ^ { 2 } - 90 x + 324 = 0$$

x uchun yechish (complex solution)

$$x=\frac{1+\sqrt{35}i}{5}\approx 0,2+1,183215957i$$

$$x=\frac{-\sqrt{35}i+1}{5}\approx 0,2-1,183215957i$$

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.

$$225x^{2}-90x+324=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 225 ni a, -90 ni b va 324 ni c bilan almashtiring.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 225\times 324}}{2\times 225}$$

-90 kvadratini chiqarish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 225\times 324}}{2\times 225}$$

-4 ni 225 marotabaga ko'paytirish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-900\times 324}}{2\times 225}$$

-900 ni 324 marotabaga ko'paytirish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-291600}}{2\times 225}$$

8100 ni -291600 ga qo'shish.

$$x=\frac{-\left(-90\right)±\sqrt{-283500}}{2\times 225}$$

-283500 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-90\right)±90\sqrt{35}i}{2\times 225}$$

-90 ning teskarisi 90 ga teng.

$$x=\frac{90±90\sqrt{35}i}{2\times 225}$$

2 ni 225 marotabaga ko'paytirish.

$$x=\frac{90±90\sqrt{35}i}{450}$$

x=\frac{90±90\sqrt{35}i}{450} tenglamasini yeching, bunda ± musbat. 90 ni 90i\sqrt{35} ga qo'shish.

$$x=\frac{90+90\sqrt{35}i}{450}$$

90+90i\sqrt{35} ni 450 ga bo'lish.

$$x=\frac{1+\sqrt{35}i}{5}$$

x=\frac{90±90\sqrt{35}i}{450} tenglamasini yeching, bunda ± manfiy. 90 dan 90i\sqrt{35} ni ayirish.

$$x=\frac{-90\sqrt{35}i+90}{450}$$

90-90i\sqrt{35} ni 450 ga bo'lish.

$$x=\frac{-\sqrt{35}i+1}{5}$$

Tenglama yechildi.

$$x=\frac{1+\sqrt{35}i}{5} x=\frac{-\sqrt{35}i+1}{5}$$

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.

$$225x^{2}-90x+324=0$$

Tenglamaning ikkala tarafidan 324 ni ayirish.

$$225x^{2}-90x+324-324=-324$$

O‘zidan 324 ayirilsa 0 qoladi.

$$225x^{2}-90x=-324$$

Ikki tarafini 225 ga bo‘ling.

$$\frac{225x^{2}-90x}{225}=-\frac{324}{225}$$

225 ga bo'lish 225 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{90}{225}\right)x=-\frac{324}{225}$$

\frac{-90}{225} ulushini 45 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{2}{5}x=-\frac{324}{225}$$

\frac{-324}{225} ulushini 9 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{2}{5}x=-\frac{36}{25}$$

-\frac{2}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{5} olish uchun. Keyin, -\frac{1}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=-\frac{36}{25}+\left(-\frac{1}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{5} kvadratini chiqarish.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{-36+1}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{36}{25} ni \frac{1}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=-\frac{7}{5}$$

x^{2}-\frac{2}{5}x+\frac{1}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{5}\right)^{2}=-\frac{7}{5}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{-\frac{7}{5}}$$

Qisqartirish.

$$x-\frac{1}{5}=\frac{\sqrt{35}i}{5} x-\frac{1}{5}=-\frac{\sqrt{35}i}{5}$$

\frac{1}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{35}i}{5} x=\frac{-\sqrt{35}i+1}{5}$$