3x+3=x^2+4x+5 eching

$$3x+3= { x }^{ 2 } +4x+5$$

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{7}i-1}{2}\approx -0,5-1,322875656i$$

$$x=\frac{-1+\sqrt{7}i}{2}\approx -0,5+1,322875656i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan x^{2} ni ayirish.

$$3x+3-x^{2}=4x+5$$

Ikkala tarafdan 4x ni ayirish.

$$3x+3-x^{2}-4x=5$$

-x ni olish uchun 3x va -4x ni birlashtirish.

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

Ikkala tarafdan 5 ni ayirish.

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

-2 olish uchun 3 dan 5 ni ayirish.

$$-x-2-x^{2}=0$$

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.

$$-x^{2}-x-2=0$$

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

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

-4 ni -1 marotabaga ko'paytirish.

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

4 ni -2 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-8}}{2\left(-1\right)}$$

1 ni -8 ga qo'shish.

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

-7 ning kvadrat ildizini chiqarish.

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

-1 ning teskarisi 1 ga teng.

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

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{7}i}{-2}$$

x=\frac{1±\sqrt{7}i}{-2} tenglamasini yeching, bunda ± musbat. 1 ni i\sqrt{7} ga qo'shish.

$$x=\frac{1+\sqrt{7}i}{-2}$$

1+i\sqrt{7} ni -2 ga bo'lish.

$$x=\frac{-\sqrt{7}i-1}{2}$$

x=\frac{1±\sqrt{7}i}{-2} tenglamasini yeching, bunda ± manfiy. 1 dan i\sqrt{7} ni ayirish.

$$x=\frac{-\sqrt{7}i+1}{-2}$$

1-i\sqrt{7} ni -2 ga bo'lish.

$$x=\frac{-1+\sqrt{7}i}{2}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{7}i-1}{2} x=\frac{-1+\sqrt{7}i}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan x^{2} ni ayirish.

$$3x+3-x^{2}=4x+5$$

Ikkala tarafdan 4x ni ayirish.

$$3x+3-x^{2}-4x=5$$

-x ni olish uchun 3x va -4x ni birlashtirish.

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

Ikkala tarafdan 3 ni ayirish.

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

2 olish uchun 5 dan 3 ni ayirish.

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

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

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

Ikki tarafini -1 ga bo‘ling.

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

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

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

-1 ni -1 ga bo'lish.

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

2 ni -1 ga bo'lish.

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

1 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{2} olish uchun. Keyin, \frac{1}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

$$x^{2}+x+\frac{1}{4}=-2+\frac{1}{4}$$

-2 ni \frac{1}{4} ga qo'shish.

$$x^{2}+x+\frac{1}{4}=-\frac{7}{4}$$

x^{2}+x+\frac{1}{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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x+\frac{1}{2}=\frac{\sqrt{7}i}{2} x+\frac{1}{2}=-\frac{\sqrt{7}i}{2}$$

Tenglamaning ikkala tarafidan \frac{1}{2} ni ayirish.

$$x=\frac{-1+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i-1}{2}$$