2x^2-14x+25=0 eching

$$2 { x }^{ 2 } -14x+25=0$$

x uchun yechish (complex solution)

$$x=\frac{7}{2}+\frac{1}{2}i=3,5+0,5i$$

$$x=\frac{7}{2}-\frac{1}{2}i=3,5-0,5i$$

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.

$$2x^{2}-14x+25=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -14 ni b va 25 ni c bilan almashtiring.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 2\times 25}}{2\times 2}$$

-14 kvadratini chiqarish.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 2\times 25}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-8\times 25}}{2\times 2}$$

-8 ni 25 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-200}}{2\times 2}$$

196 ni -200 ga qo'shish.

$$x=\frac{-\left(-14\right)±\sqrt{-4}}{2\times 2}$$

-4 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-14\right)±2i}{2\times 2}$$

-14 ning teskarisi 14 ga teng.

$$x=\frac{14±2i}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{14±2i}{4}$$

x=\frac{14±2i}{4} tenglamasini yeching, bunda ± musbat. 14 ni 2i ga qo'shish.

$$x=\frac{14+2i}{4}$$

14+2i ni 4 ga bo'lish.

$$x=\frac{7}{2}+\frac{1}{2}i$$

x=\frac{14±2i}{4} tenglamasini yeching, bunda ± manfiy. 14 dan 2i ni ayirish.

$$x=\frac{14-2i}{4}$$

14-2i ni 4 ga bo'lish.

$$x=\frac{7}{2}-\frac{1}{2}i$$

Tenglama yechildi.

$$x=\frac{7}{2}+\frac{1}{2}i x=\frac{7}{2}-\frac{1}{2}i$$

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.

$$2x^{2}-14x+25=0$$

Tenglamaning ikkala tarafidan 25 ni ayirish.

$$2x^{2}-14x+25-25=-25$$

O‘zidan 25 ayirilsa 0 qoladi.

$$2x^{2}-14x=-25$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-14x}{2}=-\frac{25}{2}$$

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

$$x^{2}+\left(-\frac{14}{2}\right)x=-\frac{25}{2}$$

-14 ni 2 ga bo'lish.

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

-7 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{2} olish uchun. Keyin, -\frac{7}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-\frac{25}{2}+\left(-\frac{7}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{2} kvadratini chiqarish.

$$x^{2}-7x+\frac{49}{4}=-\frac{25}{2}+\frac{49}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{25}{2} ni \frac{49}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-7x+\frac{49}{4}=-\frac{1}{4}$$

x^{2}-7x+\frac{49}{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.

$$\left(x-\frac{7}{2}\right)^{2}=-\frac{1}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{-\frac{1}{4}}$$

Qisqartirish.

$$x-\frac{7}{2}=\frac{1}{2}i x-\frac{7}{2}=-\frac{1}{2}i$$

\frac{7}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{7}{2}+\frac{1}{2}i x=\frac{7}{2}-\frac{1}{2}i$$