2x^2-55x+3=0 eching

$$2 { x }^{ 2 } -55x+3=0$$

x uchun yechish

$$x = \frac{\sqrt{3001} + 55}{4} \approx 27,445345925$$

$$x=\frac{55-\sqrt{3001}}{4}\approx 0,054654075$$

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.

$$2x^{2}-55x+3=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -55 ni b va 3 ni c bilan almashtiring.

$$x=\frac{-\left(-55\right)±\sqrt{\left(-55\right)^{2}-4\times 2\times 3}}{2\times 2}$$

-55 kvadratini chiqarish.

$$x=\frac{-\left(-55\right)±\sqrt{3025-4\times 2\times 3}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-55\right)±\sqrt{3025-8\times 3}}{2\times 2}$$

-8 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-55\right)±\sqrt{3025-24}}{2\times 2}$$

3025 ni -24 ga qo'shish.

$$x=\frac{-\left(-55\right)±\sqrt{3001}}{2\times 2}$$

-55 ning teskarisi 55 ga teng.

$$x=\frac{55±\sqrt{3001}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{55±\sqrt{3001}}{4}$$

x=\frac{55±\sqrt{3001}}{4} tenglamasini yeching, bunda ± musbat. 55 ni \sqrt{3001} ga qo'shish.

$$x=\frac{\sqrt{3001}+55}{4}$$

x=\frac{55±\sqrt{3001}}{4} tenglamasini yeching, bunda ± manfiy. 55 dan \sqrt{3001} ni ayirish.

$$x=\frac{55-\sqrt{3001}}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{3001}+55}{4} x=\frac{55-\sqrt{3001}}{4}$$

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.

$$2x^{2}-55x+3=0$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$2x^{2}-55x+3-3=-3$$

O‘zidan 3 ayirilsa 0 qoladi.

$$2x^{2}-55x=-3$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-55x}{2}=-\frac{3}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{55}{2}x=-\frac{3}{2}$$

-\frac{55}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{55}{4} olish uchun. Keyin, -\frac{55}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{55}{2}x+\left(-\frac{55}{4}\right)^{2}=-\frac{3}{2}+\left(-\frac{55}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{55}{4} kvadratini chiqarish.

$$x^{2}-\frac{55}{2}x+\frac{3025}{16}=-\frac{3}{2}+\frac{3025}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{3}{2} ni \frac{3025}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{55}{2}x+\frac{3025}{16}=\frac{3001}{16}$$

x^{2}-\frac{55}{2}x+\frac{3025}{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.

$$\left(x-\frac{55}{4}\right)^{2}=\frac{3001}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{55}{4}\right)^{2}}=\sqrt{\frac{3001}{16}}$$

Qisqartirish.

$$x-\frac{55}{4}=\frac{\sqrt{3001}}{4} x-\frac{55}{4}=-\frac{\sqrt{3001}}{4}$$

\frac{55}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{3001}+55}{4} x=\frac{55-\sqrt{3001}}{4}$$