8x^2-7x+2=0 eching

$$8 { x }^{ 2 } -7x+2=0$$

x uchun yechish (complex solution)

$$x=\frac{7+\sqrt{15}i}{16}\approx 0,4375+0,242061459i$$

$$x=\frac{-\sqrt{15}i+7}{16}\approx 0,4375-0,242061459i$$

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.

$$8x^{2}-7x+2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, -7 ni b va 2 ni c bilan almashtiring.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 8\times 2}}{2\times 8}$$

-7 kvadratini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 8\times 2}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-32\times 2}}{2\times 8}$$

-32 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-7\right)±\sqrt{49-64}}{2\times 8}$$

49 ni -64 ga qo'shish.

$$x=\frac{-\left(-7\right)±\sqrt{-15}}{2\times 8}$$

-15 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-7\right)±\sqrt{15}i}{2\times 8}$$

-7 ning teskarisi 7 ga teng.

$$x=\frac{7±\sqrt{15}i}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$x=\frac{7±\sqrt{15}i}{16}$$

x=\frac{7±\sqrt{15}i}{16} tenglamasini yeching, bunda ± musbat. 7 ni i\sqrt{15} ga qo'shish.

$$x=\frac{7+\sqrt{15}i}{16}$$

x=\frac{7±\sqrt{15}i}{16} tenglamasini yeching, bunda ± manfiy. 7 dan i\sqrt{15} ni ayirish.

$$x=\frac{-\sqrt{15}i+7}{16}$$

Tenglama yechildi.

$$x=\frac{7+\sqrt{15}i}{16} x=\frac{-\sqrt{15}i+7}{16}$$

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.

$$8x^{2}-7x+2=0$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$8x^{2}-7x+2-2=-2$$

O‘zidan 2 ayirilsa 0 qoladi.

$$8x^{2}-7x=-2$$

Ikki tarafini 8 ga bo‘ling.

$$\frac{8x^{2}-7x}{8}=-\frac{2}{8}$$

8 ga bo'lish 8 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{7}{8}x=-\frac{2}{8}$$

\frac{-2}{8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{7}{8}x=-\frac{1}{4}$$

-\frac{7}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{16} olish uchun. Keyin, -\frac{7}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{8}x+\left(-\frac{7}{16}\right)^{2}=-\frac{1}{4}+\left(-\frac{7}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{16} kvadratini chiqarish.

$$x^{2}-\frac{7}{8}x+\frac{49}{256}=-\frac{1}{4}+\frac{49}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{4} ni \frac{49}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{8}x+\frac{49}{256}=-\frac{15}{256}$$

x^{2}-\frac{7}{8}x+\frac{49}{256} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{7}{16}\right)^{2}=-\frac{15}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{16}\right)^{2}}=\sqrt{-\frac{15}{256}}$$

Qisqartirish.

$$x-\frac{7}{16}=\frac{\sqrt{15}i}{16} x-\frac{7}{16}=-\frac{\sqrt{15}i}{16}$$

\frac{7}{16} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{7+\sqrt{15}i}{16} x=\frac{-\sqrt{15}i+7}{16}$$