4x^2-18x+5=0 eching

$$4 x ^ { 2 } - 18 x + 5 = 0$$

x uchun yechish

$$x = \frac{\sqrt{61} + 9}{4} \approx 4,202562419$$

$$x=\frac{9-\sqrt{61}}{4}\approx 0,297437581$$

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.

$$4x^{2}-18x+5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -18 ni b va 5 ni c bilan almashtiring.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 4\times 5}}{2\times 4}$$

-18 kvadratini chiqarish.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 4\times 5}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-18\right)±\sqrt{324-16\times 5}}{2\times 4}$$

-16 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-18\right)±\sqrt{324-80}}{2\times 4}$$

324 ni -80 ga qo'shish.

$$x=\frac{-\left(-18\right)±\sqrt{244}}{2\times 4}$$

244 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-18\right)±2\sqrt{61}}{2\times 4}$$

-18 ning teskarisi 18 ga teng.

$$x=\frac{18±2\sqrt{61}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{18±2\sqrt{61}}{8}$$

x=\frac{18±2\sqrt{61}}{8} tenglamasini yeching, bunda ± musbat. 18 ni 2\sqrt{61} ga qo'shish.

$$x=\frac{2\sqrt{61}+18}{8}$$

18+2\sqrt{61} ni 8 ga bo'lish.

$$x=\frac{\sqrt{61}+9}{4}$$

x=\frac{18±2\sqrt{61}}{8} tenglamasini yeching, bunda ± manfiy. 18 dan 2\sqrt{61} ni ayirish.

$$x=\frac{18-2\sqrt{61}}{8}$$

18-2\sqrt{61} ni 8 ga bo'lish.

$$x=\frac{9-\sqrt{61}}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{61}+9}{4} x=\frac{9-\sqrt{61}}{4}$$

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.

$$4x^{2}-18x+5=0$$

Tenglamaning ikkala tarafidan 5 ni ayirish.

$$4x^{2}-18x+5-5=-5$$

O‘zidan 5 ayirilsa 0 qoladi.

$$4x^{2}-18x=-5$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-18x}{4}=-\frac{5}{4}$$

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

$$x^{2}+\left(-\frac{18}{4}\right)x=-\frac{5}{4}$$

\frac{-18}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{9}{2}x=-\frac{5}{4}$$

-\frac{9}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{4} olish uchun. Keyin, -\frac{9}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=-\frac{5}{4}+\left(-\frac{9}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{4} kvadratini chiqarish.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-\frac{5}{4}+\frac{81}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{5}{4} ni \frac{81}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{61}{16}$$

x^{2}-\frac{9}{2}x+\frac{81}{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.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{61}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{61}{16}}$$

Qisqartirish.

$$x-\frac{9}{4}=\frac{\sqrt{61}}{4} x-\frac{9}{4}=-\frac{\sqrt{61}}{4}$$

\frac{9}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{61}+9}{4} x=\frac{9-\sqrt{61}}{4}$$