4x^2-63x+270=0 eching

$$4 { x }^{ 2 } -63x+270=0$$

x uchun yechish (complex solution)

$$x=\frac{63+3\sqrt{39}i}{8}\approx 7,875+2,341874249i$$

$$x=\frac{-3\sqrt{39}i+63}{8}\approx 7,875-2,341874249i$$

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.

$$4x^{2}-63x+270=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -63 ni b va 270 ni c bilan almashtiring.

$$x=\frac{-\left(-63\right)±\sqrt{\left(-63\right)^{2}-4\times 4\times 270}}{2\times 4}$$

-63 kvadratini chiqarish.

$$x=\frac{-\left(-63\right)±\sqrt{3969-4\times 4\times 270}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-63\right)±\sqrt{3969-16\times 270}}{2\times 4}$$

-16 ni 270 marotabaga ko'paytirish.

$$x=\frac{-\left(-63\right)±\sqrt{3969-4320}}{2\times 4}$$

3969 ni -4320 ga qo'shish.

$$x=\frac{-\left(-63\right)±\sqrt{-351}}{2\times 4}$$

-351 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-63\right)±3\sqrt{39}i}{2\times 4}$$

-63 ning teskarisi 63 ga teng.

$$x=\frac{63±3\sqrt{39}i}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{63±3\sqrt{39}i}{8}$$

x=\frac{63±3\sqrt{39}i}{8} tenglamasini yeching, bunda ± musbat. 63 ni 3i\sqrt{39} ga qo'shish.

$$x=\frac{63+3\sqrt{39}i}{8}$$

x=\frac{63±3\sqrt{39}i}{8} tenglamasini yeching, bunda ± manfiy. 63 dan 3i\sqrt{39} ni ayirish.

$$x=\frac{-3\sqrt{39}i+63}{8}$$

Tenglama yechildi.

$$x=\frac{63+3\sqrt{39}i}{8} x=\frac{-3\sqrt{39}i+63}{8}$$

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.

$$4x^{2}-63x+270=0$$

Tenglamaning ikkala tarafidan 270 ni ayirish.

$$4x^{2}-63x+270-270=-270$$

O‘zidan 270 ayirilsa 0 qoladi.

$$4x^{2}-63x=-270$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-63x}{4}=-\frac{270}{4}$$

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

$$x^{2}-\frac{63}{4}x=-\frac{270}{4}$$

\frac{-270}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{63}{4}x=-\frac{135}{2}$$

-\frac{63}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{63}{8} olish uchun. Keyin, -\frac{63}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{63}{4}x+\left(-\frac{63}{8}\right)^{2}=-\frac{135}{2}+\left(-\frac{63}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{63}{8} kvadratini chiqarish.

$$x^{2}-\frac{63}{4}x+\frac{3969}{64}=-\frac{135}{2}+\frac{3969}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{135}{2} ni \frac{3969}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{63}{4}x+\frac{3969}{64}=-\frac{351}{64}$$

x^{2}-\frac{63}{4}x+\frac{3969}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{63}{8}\right)^{2}=-\frac{351}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{63}{8}\right)^{2}}=\sqrt{-\frac{351}{64}}$$

Qisqartirish.

$$x-\frac{63}{8}=\frac{3\sqrt{39}i}{8} x-\frac{63}{8}=-\frac{3\sqrt{39}i}{8}$$

\frac{63}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{63+3\sqrt{39}i}{8} x=\frac{-3\sqrt{39}i+63}{8}$$