5-x^2+3x=12 eching

$$5- { x }^{ 2 } +3x=12$$

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{19}i+3}{2}\approx 1,5-2,179449472i$$

$$x=\frac{3+\sqrt{19}i}{2}\approx 1,5+2,179449472i$$

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.

$$-x^{2}+3x+5=12$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$-x^{2}+3x+5-12=12-12$$

O‘zidan 12 ayirilsa 0 qoladi.

$$-x^{2}+3x+5-12=0$$

5 dan 12 ni ayirish.

$$-x^{2}+3x-7=0$$

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 -7 ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+4\left(-7\right)}}{2\left(-1\right)}$$

4 ni -7 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9-28}}{2\left(-1\right)}$$

9 ni -28 ga qo'shish.

$$x=\frac{-3±\sqrt{-19}}{2\left(-1\right)}$$

-19 ning kvadrat ildizini chiqarish.

$$x=\frac{-3±\sqrt{19}i}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{19}i}{-2}$$

x=\frac{-3±\sqrt{19}i}{-2} tenglamasini yeching, bunda ± musbat. -3 ni i\sqrt{19} ga qo'shish.

$$x=\frac{-3+\sqrt{19}i}{-2}$$

-3+i\sqrt{19} ni -2 ga bo'lish.

$$x=\frac{-\sqrt{19}i+3}{2}$$

x=\frac{-3±\sqrt{19}i}{-2} tenglamasini yeching, bunda ± manfiy. -3 dan i\sqrt{19} ni ayirish.

$$x=\frac{-\sqrt{19}i-3}{-2}$$

-3-i\sqrt{19} ni -2 ga bo'lish.

$$x=\frac{3+\sqrt{19}i}{2}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{19}i+3}{2} x=\frac{3+\sqrt{19}i}{2}$$

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.

$$-x^{2}+3x+5=12$$

Tenglamaning ikkala tarafidan 5 ni ayirish.

$$-x^{2}+3x+5-5=12-5$$

O‘zidan 5 ayirilsa 0 qoladi.

$$-x^{2}+3x=12-5$$

12 dan 5 ni ayirish.

$$-x^{2}+3x=7$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+3x}{-1}=\frac{7}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{3}{-1}x=\frac{7}{-1}$$

3 ni -1 ga bo'lish.

$$x^{2}-3x=\frac{7}{-1}$$

7 ni -1 ga bo'lish.

$$x^{2}-3x=-7$$

-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.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-7+\left(-\frac{3}{2}\right)^{2}$$

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

$$x^{2}-3x+\frac{9}{4}=-7+\frac{9}{4}$$

-7 ni \frac{9}{4} ga qo'shish.

$$x^{2}-3x+\frac{9}{4}=-\frac{19}{4}$$

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.

$$\left(x-\frac{3}{2}\right)^{2}=-\frac{19}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{-\frac{19}{4}}$$

Qisqartirish.

$$x-\frac{3}{2}=\frac{\sqrt{19}i}{2} x-\frac{3}{2}=-\frac{\sqrt{19}i}{2}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3+\sqrt{19}i}{2} x=\frac{-\sqrt{19}i+3}{2}$$