x^2+3x+54=0 eching

$${ x }^{ 2 } +3x+54 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-3+3\sqrt{23}i}{2}\approx -1,5+7,193747285i$$

$$x=\frac{-3\sqrt{23}i-3}{2}\approx -1,5-7,193747285i$$

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.

$$x^{2}+3x+54=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 3 ni b va 54 ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}-4\times 54}}{2}$$

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\times 54}}{2}$$

-4 ni 54 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9-216}}{2}$$

9 ni -216 ga qo'shish.

$$x=\frac{-3±\sqrt{-207}}{2}$$

-207 ning kvadrat ildizini chiqarish.

$$x=\frac{-3±3\sqrt{23}i}{2}$$

x=\frac{-3±3\sqrt{23}i}{2} tenglamasini yeching, bunda ± musbat. -3 ni 3i\sqrt{23} ga qo'shish.

$$x=\frac{-3+3\sqrt{23}i}{2}$$

x=\frac{-3±3\sqrt{23}i}{2} tenglamasini yeching, bunda ± manfiy. -3 dan 3i\sqrt{23} ni ayirish.

$$x=\frac{-3\sqrt{23}i-3}{2}$$

Tenglama yechildi.

$$x=\frac{-3+3\sqrt{23}i}{2} x=\frac{-3\sqrt{23}i-3}{2}$$

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.

$$x^{2}+3x+54=0$$

Tenglamaning ikkala tarafidan 54 ni ayirish.

$$x^{2}+3x+54-54=-54$$

O‘zidan 54 ayirilsa 0 qoladi.

$$x^{2}+3x=-54$$

3 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{2} olish uchun. Keyin, \frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=-54+\left(\frac{3}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{2} kvadratini chiqarish.

$$x^{2}+3x+\frac{9}{4}=-54+\frac{9}{4}$$

-54 ni \frac{9}{4} ga qo'shish.

$$x^{2}+3x+\frac{9}{4}=-\frac{207}{4}$$

x^{2}+3x+\frac{9}{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.

$$\left(x+\frac{3}{2}\right)^{2}=-\frac{207}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{-\frac{207}{4}}$$

Qisqartirish.

$$x+\frac{3}{2}=\frac{3\sqrt{23}i}{2} x+\frac{3}{2}=-\frac{3\sqrt{23}i}{2}$$

Tenglamaning ikkala tarafidan \frac{3}{2} ni ayirish.

$$x=\frac{-3+3\sqrt{23}i}{2} x=\frac{-3\sqrt{23}i-3}{2}$$