x-x^2=3 eching

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

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{11}i+1}{2}\approx 0,5-1,658312395i$$

$$x=\frac{1+\sqrt{11}i}{2}\approx 0,5+1,658312395i$$

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.

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

Tenglamaning ikkala tarafidan 3 ni ayirish.

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

O‘zidan 3 ayirilsa 0 qoladi.

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

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 -3 ni c bilan almashtiring.

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

1 kvadratini chiqarish.

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

-4 ni -1 marotabaga ko'paytirish.

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

4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{1-12}}{2\left(-1\right)}$$

1 ni -12 ga qo'shish.

$$x=\frac{-1±\sqrt{-11}}{2\left(-1\right)}$$

-11 ning kvadrat ildizini chiqarish.

$$x=\frac{-1±\sqrt{11}i}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-1±\sqrt{11}i}{-2}$$

x=\frac{-1±\sqrt{11}i}{-2} tenglamasini yeching, bunda ± musbat. -1 ni i\sqrt{11} ga qo'shish.

$$x=\frac{-1+\sqrt{11}i}{-2}$$

-1+i\sqrt{11} ni -2 ga bo'lish.

$$x=\frac{-\sqrt{11}i+1}{2}$$

x=\frac{-1±\sqrt{11}i}{-2} tenglamasini yeching, bunda ± manfiy. -1 dan i\sqrt{11} ni ayirish.

$$x=\frac{-\sqrt{11}i-1}{-2}$$

-1-i\sqrt{11} ni -2 ga bo'lish.

$$x=\frac{1+\sqrt{11}i}{2}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{11}i+1}{2} x=\frac{1+\sqrt{11}i}{2}$$

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.

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

Ikki tarafini -1 ga bo‘ling.

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

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

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

1 ni -1 ga bo'lish.

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

3 ni -1 ga bo'lish.

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

-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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-3+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=-3+\frac{1}{4}$$

-3 ni \frac{1}{4} ga qo'shish.

$$x^{2}-x+\frac{1}{4}=-\frac{11}{4}$$

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.

$$\left(x-\frac{1}{2}\right)^{2}=-\frac{11}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{-\frac{11}{4}}$$

Qisqartirish.

$$x-\frac{1}{2}=\frac{\sqrt{11}i}{2} x-\frac{1}{2}=-\frac{\sqrt{11}i}{2}$$

\frac{1}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{1+\sqrt{11}i}{2} x=\frac{-\sqrt{11}i+1}{2}$$