25x^2-90x+87=0 eching

$$25 { x }^{ 2 } -90x+87=0$$

x uchun yechish (complex solution)

$$x=\frac{9+\sqrt{6}i}{5}\approx 1,8+0,489897949i$$

$$x=\frac{-\sqrt{6}i+9}{5}\approx 1,8-0,489897949i$$

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.

$$25x^{2}-90x+87=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 25 ni a, -90 ni b va 87 ni c bilan almashtiring.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 25\times 87}}{2\times 25}$$

-90 kvadratini chiqarish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 25\times 87}}{2\times 25}$$

-4 ni 25 marotabaga ko'paytirish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-100\times 87}}{2\times 25}$$

-100 ni 87 marotabaga ko'paytirish.

$$x=\frac{-\left(-90\right)±\sqrt{8100-8700}}{2\times 25}$$

8100 ni -8700 ga qo'shish.

$$x=\frac{-\left(-90\right)±\sqrt{-600}}{2\times 25}$$

-600 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-90\right)±10\sqrt{6}i}{2\times 25}$$

-90 ning teskarisi 90 ga teng.

$$x=\frac{90±10\sqrt{6}i}{2\times 25}$$

2 ni 25 marotabaga ko'paytirish.

$$x=\frac{90±10\sqrt{6}i}{50}$$

x=\frac{90±10\sqrt{6}i}{50} tenglamasini yeching, bunda ± musbat. 90 ni 10i\sqrt{6} ga qo'shish.

$$x=\frac{90+10\sqrt{6}i}{50}$$

90+10i\sqrt{6} ni 50 ga bo'lish.

$$x=\frac{9+\sqrt{6}i}{5}$$

x=\frac{90±10\sqrt{6}i}{50} tenglamasini yeching, bunda ± manfiy. 90 dan 10i\sqrt{6} ni ayirish.

$$x=\frac{-10\sqrt{6}i+90}{50}$$

90-10i\sqrt{6} ni 50 ga bo'lish.

$$x=\frac{-\sqrt{6}i+9}{5}$$

Tenglama yechildi.

$$x=\frac{9+\sqrt{6}i}{5} x=\frac{-\sqrt{6}i+9}{5}$$

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.

$$25x^{2}-90x+87=0$$

Tenglamaning ikkala tarafidan 87 ni ayirish.

$$25x^{2}-90x+87-87=-87$$

O‘zidan 87 ayirilsa 0 qoladi.

$$25x^{2}-90x=-87$$

Ikki tarafini 25 ga bo‘ling.

$$\frac{25x^{2}-90x}{25}=-\frac{87}{25}$$

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

$$x^{2}+\left(-\frac{90}{25}\right)x=-\frac{87}{25}$$

\frac{-90}{25} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{18}{5}x=-\frac{87}{25}$$

-\frac{18}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{5} olish uchun. Keyin, -\frac{9}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{18}{5}x+\left(-\frac{9}{5}\right)^{2}=-\frac{87}{25}+\left(-\frac{9}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{5} kvadratini chiqarish.

$$x^{2}-\frac{18}{5}x+\frac{81}{25}=\frac{-87+81}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{87}{25} ni \frac{81}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{18}{5}x+\frac{81}{25}=-\frac{6}{25}$$

x^{2}-\frac{18}{5}x+\frac{81}{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.

$$\left(x-\frac{9}{5}\right)^{2}=-\frac{6}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{5}\right)^{2}}=\sqrt{-\frac{6}{25}}$$

Qisqartirish.

$$x-\frac{9}{5}=\frac{\sqrt{6}i}{5} x-\frac{9}{5}=-\frac{\sqrt{6}i}{5}$$

\frac{9}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{9+\sqrt{6}i}{5} x=\frac{-\sqrt{6}i+9}{5}$$