4x^2-2x+9=0 eching

$$4 x ^ { 2 } - 2 x + 9 = 0$$

x uchun yechish (complex solution)

$$x=\frac{1+\sqrt{35}i}{4}\approx 0,25+1,479019946i$$

$$x=\frac{-\sqrt{35}i+1}{4}\approx 0,25-1,479019946i$$

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.

$$4x^{2}-2x+9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -2 ni b va 9 ni c bilan almashtiring.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 4\times 9}}{2\times 4}$$

-2 kvadratini chiqarish.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 4\times 9}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-16\times 9}}{2\times 4}$$

-16 ni 9 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4-144}}{2\times 4}$$

4 ni -144 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{-140}}{2\times 4}$$

-140 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±2\sqrt{35}i}{2\times 4}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±2\sqrt{35}i}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{2±2\sqrt{35}i}{8}$$

x=\frac{2±2\sqrt{35}i}{8} tenglamasini yeching, bunda ± musbat. 2 ni 2i\sqrt{35} ga qo'shish.

$$x=\frac{2+2\sqrt{35}i}{8}$$

2+2i\sqrt{35} ni 8 ga bo'lish.

$$x=\frac{1+\sqrt{35}i}{4}$$

x=\frac{2±2\sqrt{35}i}{8} tenglamasini yeching, bunda ± manfiy. 2 dan 2i\sqrt{35} ni ayirish.

$$x=\frac{-2\sqrt{35}i+2}{8}$$

2-2i\sqrt{35} ni 8 ga bo'lish.

$$x=\frac{-\sqrt{35}i+1}{4}$$

Tenglama yechildi.

$$x=\frac{1+\sqrt{35}i}{4} x=\frac{-\sqrt{35}i+1}{4}$$

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.

$$4x^{2}-2x+9=0$$

Tenglamaning ikkala tarafidan 9 ni ayirish.

$$4x^{2}-2x+9-9=-9$$

O‘zidan 9 ayirilsa 0 qoladi.

$$4x^{2}-2x=-9$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-2x}{4}=-\frac{9}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{2}{4}\right)x=-\frac{9}{4}$$

\frac{-2}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{2}x=-\frac{9}{4}$$

-\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{4} olish uchun. Keyin, -\frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=-\frac{9}{4}+\left(-\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{4} kvadratini chiqarish.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{9}{4}+\frac{1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{9}{4} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{35}{16}$$

x^{2}-\frac{1}{2}x+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{35}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{35}{16}}$$

Qisqartirish.

$$x-\frac{1}{4}=\frac{\sqrt{35}i}{4} x-\frac{1}{4}=-\frac{\sqrt{35}i}{4}$$

\frac{1}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{35}i}{4} x=\frac{-\sqrt{35}i+1}{4}$$