x^2-15x+100=0 eching

$$x ^ { 2 } - 15 x + 100 = 0$$

x uchun yechish (complex solution)

$$x=\frac{15+5\sqrt{7}i}{2}\approx 7,5+6,614378278i$$

$$x=\frac{-5\sqrt{7}i+15}{2}\approx 7,5-6,614378278i$$

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.

$$x^{2}-15x+100=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -15 ni b va 100 ni c bilan almashtiring.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 100}}{2}$$

-15 kvadratini chiqarish.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 100}}{2}$$

-4 ni 100 marotabaga ko'paytirish.

$$x=\frac{-\left(-15\right)±\sqrt{225-400}}{2}$$

225 ni -400 ga qo'shish.

$$x=\frac{-\left(-15\right)±\sqrt{-175}}{2}$$

-175 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-15\right)±5\sqrt{7}i}{2}$$

-15 ning teskarisi 15 ga teng.

$$x=\frac{15±5\sqrt{7}i}{2}$$

x=\frac{15±5\sqrt{7}i}{2} tenglamasini yeching, bunda ± musbat. 15 ni 5i\sqrt{7} ga qo'shish.

$$x=\frac{15+5\sqrt{7}i}{2}$$

x=\frac{15±5\sqrt{7}i}{2} tenglamasini yeching, bunda ± manfiy. 15 dan 5i\sqrt{7} ni ayirish.

$$x=\frac{-5\sqrt{7}i+15}{2}$$

Tenglama yechildi.

$$x=\frac{15+5\sqrt{7}i}{2} x=\frac{-5\sqrt{7}i+15}{2}$$

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.

$$x^{2}-15x+100=0$$

Tenglamaning ikkala tarafidan 100 ni ayirish.

$$x^{2}-15x+100-100=-100$$

O‘zidan 100 ayirilsa 0 qoladi.

$$x^{2}-15x=-100$$

-15 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{15}{2} olish uchun. Keyin, -\frac{15}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-100+\left(-\frac{15}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{15}{2} kvadratini chiqarish.

$$x^{2}-15x+\frac{225}{4}=-100+\frac{225}{4}$$

-100 ni \frac{225}{4} ga qo'shish.

$$x^{2}-15x+\frac{225}{4}=-\frac{175}{4}$$

x^{2}-15x+\frac{225}{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.

$$\left(x-\frac{15}{2}\right)^{2}=-\frac{175}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{-\frac{175}{4}}$$

Qisqartirish.

$$x-\frac{15}{2}=\frac{5\sqrt{7}i}{2} x-\frac{15}{2}=-\frac{5\sqrt{7}i}{2}$$

\frac{15}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{15+5\sqrt{7}i}{2} x=\frac{-5\sqrt{7}i+15}{2}$$