2x^2-70x+1225=0 eching

$$2 x ^ { 2 } - 70 x + 1225 = 0$$

x uchun yechish (complex solution)

$$x=\frac{35}{2}+\frac{35}{2}i=17,5+17,5i$$

$$x=\frac{35}{2}-\frac{35}{2}i=17,5-17,5i$$

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.

$$2x^{2}-70x+1225=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -70 ni b va 1225 ni c bilan almashtiring.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 2\times 1225}}{2\times 2}$$

-70 kvadratini chiqarish.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 2\times 1225}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-70\right)±\sqrt{4900-8\times 1225}}{2\times 2}$$

-8 ni 1225 marotabaga ko'paytirish.

$$x=\frac{-\left(-70\right)±\sqrt{4900-9800}}{2\times 2}$$

4900 ni -9800 ga qo'shish.

$$x=\frac{-\left(-70\right)±\sqrt{-4900}}{2\times 2}$$

-4900 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-70\right)±70i}{2\times 2}$$

-70 ning teskarisi 70 ga teng.

$$x=\frac{70±70i}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{70±70i}{4}$$

x=\frac{70±70i}{4} tenglamasini yeching, bunda ± musbat. 70 ni 70i ga qo'shish.

$$x=\frac{70+70i}{4}$$

70+70i ni 4 ga bo'lish.

$$x=\frac{35}{2}+\frac{35}{2}i$$

x=\frac{70±70i}{4} tenglamasini yeching, bunda ± manfiy. 70 dan 70i ni ayirish.

$$x=\frac{70-70i}{4}$$

70-70i ni 4 ga bo'lish.

$$x=\frac{35}{2}-\frac{35}{2}i$$

Tenglama yechildi.

$$x=\frac{35}{2}+\frac{35}{2}i x=\frac{35}{2}-\frac{35}{2}i$$

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.

$$2x^{2}-70x+1225=0$$

Tenglamaning ikkala tarafidan 1225 ni ayirish.

$$2x^{2}-70x+1225-1225=-1225$$

O‘zidan 1225 ayirilsa 0 qoladi.

$$2x^{2}-70x=-1225$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-70x}{2}=-\frac{1225}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{70}{2}\right)x=-\frac{1225}{2}$$

-70 ni 2 ga bo'lish.

$$x^{2}-35x=-\frac{1225}{2}$$

-35 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{35}{2} olish uchun. Keyin, -\frac{35}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-35x+\left(-\frac{35}{2}\right)^{2}=-\frac{1225}{2}+\left(-\frac{35}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{35}{2} kvadratini chiqarish.

$$x^{2}-35x+\frac{1225}{4}=-\frac{1225}{2}+\frac{1225}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1225}{2} ni \frac{1225}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-35x+\frac{1225}{4}=-\frac{1225}{4}$$

x^{2}-35x+\frac{1225}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{35}{2}\right)^{2}=-\frac{1225}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{35}{2}\right)^{2}}=\sqrt{-\frac{1225}{4}}$$

Qisqartirish.

$$x-\frac{35}{2}=\frac{35}{2}i x-\frac{35}{2}=-\frac{35}{2}i$$

\frac{35}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{35}{2}+\frac{35}{2}i x=\frac{35}{2}-\frac{35}{2}i$$