-3x^2-24x-51=0 eching

$$- 3 x ^ { 2 } - 24 x - 51 = 0$$

x uchun yechish (complex solution)

$$x=-4+i$$

$$x=-4-i$$

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.

$$-3x^{2}-24x-51=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, -24 ni b va -51 ni c bilan almashtiring.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-3\right)\left(-51\right)}}{2\left(-3\right)}$$

-24 kvadratini chiqarish.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\left(-3\right)\left(-51\right)}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-24\right)±\sqrt{576+12\left(-51\right)}}{2\left(-3\right)}$$

12 ni -51 marotabaga ko'paytirish.

$$x=\frac{-\left(-24\right)±\sqrt{576-612}}{2\left(-3\right)}$$

576 ni -612 ga qo'shish.

$$x=\frac{-\left(-24\right)±\sqrt{-36}}{2\left(-3\right)}$$

-36 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-24\right)±6i}{2\left(-3\right)}$$

-24 ning teskarisi 24 ga teng.

$$x=\frac{24±6i}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{24±6i}{-6}$$

x=\frac{24±6i}{-6} tenglamasini yeching, bunda ± musbat. 24 ni 6i ga qo'shish.

$$x=\frac{24+6i}{-6}$$

24+6i ni -6 ga bo'lish.

$$x=-4-i$$

x=\frac{24±6i}{-6} tenglamasini yeching, bunda ± manfiy. 24 dan 6i ni ayirish.

$$x=\frac{24-6i}{-6}$$

24-6i ni -6 ga bo'lish.

$$x=-4+i$$

Tenglama yechildi.

$$x=-4-i x=-4+i$$

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.

$$-3x^{2}-24x-51=0$$

51 ni tenglamaning ikkala tarafiga qo'shish.

$$-3x^{2}-24x-51-\left(-51\right)=-\left(-51\right)$$

O‘zidan -51 ayirilsa 0 qoladi.

$$-3x^{2}-24x=-\left(-51\right)$$

0 dan -51 ni ayirish.

$$-3x^{2}-24x=51$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}-24x}{-3}=\frac{51}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{24}{-3}\right)x=\frac{51}{-3}$$

-24 ni -3 ga bo'lish.

$$x^{2}+8x=\frac{51}{-3}$$

51 ni -3 ga bo'lish.

$$x^{2}+8x=-17$$

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

$$x^{2}+8x+4^{2}=-17+4^{2}$$

4 kvadratini chiqarish.

$$x^{2}+8x+16=-17+16$$

-17 ni 16 ga qo'shish.

$$x^{2}+8x+16=-1$$

x^{2}+8x+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.

$$\left(x+4\right)^{2}=-1$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{-1}$$

Qisqartirish.

$$x+4=i x+4=-i$$

Tenglamaning ikkala tarafidan 4 ni ayirish.

$$x=-4+i x=-4-i$$