x^2-115x+4254=0 eching

$${ x }^{ 2 } -115x+4254=0$$

x uchun yechish (complex solution)

$$x=\frac{115+\sqrt{3791}i}{2}\approx 57,5+30,785548558i$$

$$x=\frac{-\sqrt{3791}i+115}{2}\approx 57,5-30,785548558i$$

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.

$$x^{2}-115x+4254=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -115 ni b va 4254 ni c bilan almashtiring.

$$x=\frac{-\left(-115\right)±\sqrt{\left(-115\right)^{2}-4\times 4254}}{2}$$

-115 kvadratini chiqarish.

$$x=\frac{-\left(-115\right)±\sqrt{13225-4\times 4254}}{2}$$

-4 ni 4254 marotabaga ko'paytirish.

$$x=\frac{-\left(-115\right)±\sqrt{13225-17016}}{2}$$

13225 ni -17016 ga qo'shish.

$$x=\frac{-\left(-115\right)±\sqrt{-3791}}{2}$$

-3791 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-115\right)±\sqrt{3791}i}{2}$$

-115 ning teskarisi 115 ga teng.

$$x=\frac{115±\sqrt{3791}i}{2}$$

x=\frac{115±\sqrt{3791}i}{2} tenglamasini yeching, bunda ± musbat. 115 ni i\sqrt{3791} ga qo'shish.

$$x=\frac{115+\sqrt{3791}i}{2}$$

x=\frac{115±\sqrt{3791}i}{2} tenglamasini yeching, bunda ± manfiy. 115 dan i\sqrt{3791} ni ayirish.

$$x=\frac{-\sqrt{3791}i+115}{2}$$

Tenglama yechildi.

$$x=\frac{115+\sqrt{3791}i}{2} x=\frac{-\sqrt{3791}i+115}{2}$$

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.

$$x^{2}-115x+4254=0$$

Tenglamaning ikkala tarafidan 4254 ni ayirish.

$$x^{2}-115x+4254-4254=-4254$$

O‘zidan 4254 ayirilsa 0 qoladi.

$$x^{2}-115x=-4254$$

-115 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{115}{2} olish uchun. Keyin, -\frac{115}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-115x+\left(-\frac{115}{2}\right)^{2}=-4254+\left(-\frac{115}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{115}{2} kvadratini chiqarish.

$$x^{2}-115x+\frac{13225}{4}=-4254+\frac{13225}{4}$$

-4254 ni \frac{13225}{4} ga qo'shish.

$$x^{2}-115x+\frac{13225}{4}=-\frac{3791}{4}$$

x^{2}-115x+\frac{13225}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{115}{2}\right)^{2}=-\frac{3791}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{115}{2}\right)^{2}}=\sqrt{-\frac{3791}{4}}$$

Qisqartirish.

$$x-\frac{115}{2}=\frac{\sqrt{3791}i}{2} x-\frac{115}{2}=-\frac{\sqrt{3791}i}{2}$$

\frac{115}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{115+\sqrt{3791}i}{2} x=\frac{-\sqrt{3791}i+115}{2}$$