4x^2-75x+50=0 eching

$$4 x ^ { 2 } - 75 x + 50 = 0$$

x uchun yechish

$$x = \frac{5 \sqrt{193} + 75}{8} \approx 18,057777493$$

$$x=\frac{75-5\sqrt{193}}{8}\approx 0,692222507$$

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.

$$4x^{2}-75x+50=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -75 ni b va 50 ni c bilan almashtiring.

$$x=\frac{-\left(-75\right)±\sqrt{\left(-75\right)^{2}-4\times 4\times 50}}{2\times 4}$$

-75 kvadratini chiqarish.

$$x=\frac{-\left(-75\right)±\sqrt{5625-4\times 4\times 50}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-75\right)±\sqrt{5625-16\times 50}}{2\times 4}$$

-16 ni 50 marotabaga ko'paytirish.

$$x=\frac{-\left(-75\right)±\sqrt{5625-800}}{2\times 4}$$

5625 ni -800 ga qo'shish.

$$x=\frac{-\left(-75\right)±\sqrt{4825}}{2\times 4}$$

4825 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-75\right)±5\sqrt{193}}{2\times 4}$$

-75 ning teskarisi 75 ga teng.

$$x=\frac{75±5\sqrt{193}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{75±5\sqrt{193}}{8}$$

x=\frac{75±5\sqrt{193}}{8} tenglamasini yeching, bunda ± musbat. 75 ni 5\sqrt{193} ga qo'shish.

$$x=\frac{5\sqrt{193}+75}{8}$$

x=\frac{75±5\sqrt{193}}{8} tenglamasini yeching, bunda ± manfiy. 75 dan 5\sqrt{193} ni ayirish.

$$x=\frac{75-5\sqrt{193}}{8}$$

Tenglama yechildi.

$$x=\frac{5\sqrt{193}+75}{8} x=\frac{75-5\sqrt{193}}{8}$$

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.

$$4x^{2}-75x+50=0$$

Tenglamaning ikkala tarafidan 50 ni ayirish.

$$4x^{2}-75x+50-50=-50$$

O‘zidan 50 ayirilsa 0 qoladi.

$$4x^{2}-75x=-50$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-75x}{4}=-\frac{50}{4}$$

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

$$x^{2}-\frac{75}{4}x=-\frac{50}{4}$$

\frac{-50}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{75}{4}x=-\frac{25}{2}$$

-\frac{75}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{75}{8} olish uchun. Keyin, -\frac{75}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{75}{4}x+\left(-\frac{75}{8}\right)^{2}=-\frac{25}{2}+\left(-\frac{75}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{75}{8} kvadratini chiqarish.

$$x^{2}-\frac{75}{4}x+\frac{5625}{64}=-\frac{25}{2}+\frac{5625}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{25}{2} ni \frac{5625}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{75}{4}x+\frac{5625}{64}=\frac{4825}{64}$$

x^{2}-\frac{75}{4}x+\frac{5625}{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.

$$\left(x-\frac{75}{8}\right)^{2}=\frac{4825}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{75}{8}\right)^{2}}=\sqrt{\frac{4825}{64}}$$

Qisqartirish.

$$x-\frac{75}{8}=\frac{5\sqrt{193}}{8} x-\frac{75}{8}=-\frac{5\sqrt{193}}{8}$$

\frac{75}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5\sqrt{193}+75}{8} x=\frac{75-5\sqrt{193}}{8}$$