x^2+19x+100=0 eching

$${ x }^{ 2 } +19x+100=0$$

x uchun yechish (complex solution)

$$x=\frac{-19+\sqrt{39}i}{2}\approx -9,5+3,122498999i$$

$$x=\frac{-\sqrt{39}i-19}{2}\approx -9,5-3,122498999i$$

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.

$$x^{2}+19x+100=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 19 ni b va 100 ni c bilan almashtiring.

$$x=\frac{-19±\sqrt{19^{2}-4\times 100}}{2}$$

19 kvadratini chiqarish.

$$x=\frac{-19±\sqrt{361-4\times 100}}{2}$$

-4 ni 100 marotabaga ko'paytirish.

$$x=\frac{-19±\sqrt{361-400}}{2}$$

361 ni -400 ga qo'shish.

$$x=\frac{-19±\sqrt{-39}}{2}$$

-39 ning kvadrat ildizini chiqarish.

$$x=\frac{-19±\sqrt{39}i}{2}$$

x=\frac{-19±\sqrt{39}i}{2} tenglamasini yeching, bunda ± musbat. -19 ni i\sqrt{39} ga qo'shish.

$$x=\frac{-19+\sqrt{39}i}{2}$$

x=\frac{-19±\sqrt{39}i}{2} tenglamasini yeching, bunda ± manfiy. -19 dan i\sqrt{39} ni ayirish.

$$x=\frac{-\sqrt{39}i-19}{2}$$

Tenglama yechildi.

$$x=\frac{-19+\sqrt{39}i}{2} x=\frac{-\sqrt{39}i-19}{2}$$

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.

$$x^{2}+19x+100=0$$

Tenglamaning ikkala tarafidan 100 ni ayirish.

$$x^{2}+19x+100-100=-100$$

O‘zidan 100 ayirilsa 0 qoladi.

$$x^{2}+19x=-100$$

19 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{19}{2} olish uchun. Keyin, \frac{19}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+19x+\left(\frac{19}{2}\right)^{2}=-100+\left(\frac{19}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{19}{2} kvadratini chiqarish.

$$x^{2}+19x+\frac{361}{4}=-100+\frac{361}{4}$$

-100 ni \frac{361}{4} ga qo'shish.

$$x^{2}+19x+\frac{361}{4}=-\frac{39}{4}$$

x^{2}+19x+\frac{361}{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.

$$\left(x+\frac{19}{2}\right)^{2}=-\frac{39}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{19}{2}\right)^{2}}=\sqrt{-\frac{39}{4}}$$

Qisqartirish.

$$x+\frac{19}{2}=\frac{\sqrt{39}i}{2} x+\frac{19}{2}=-\frac{\sqrt{39}i}{2}$$

Tenglamaning ikkala tarafidan \frac{19}{2} ni ayirish.

$$x=\frac{-19+\sqrt{39}i}{2} x=\frac{-\sqrt{39}i-19}{2}$$