-x^2-x-1=0 eching

$$- x ^ { 2 } - x - 1 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-\sqrt{3}i-1}{2}\approx -0,5-0,866025404i$$

$$x=\frac{-1+\sqrt{3}i}{2}\approx -0,5+0,866025404i$$

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.

$$-x^{2}-x-1=0$$

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 -1 ni c bilan almashtiring.

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

-4 ni -1 marotabaga ko'paytirish.

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

4 ni -1 marotabaga ko'paytirish.

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

1 ni -4 ga qo'shish.

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

-3 ning kvadrat ildizini chiqarish.

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

-1 ning teskarisi 1 ga teng.

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

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{3}i}{-2}$$

x=\frac{1±\sqrt{3}i}{-2} tenglamasini yeching, bunda ± musbat. 1 ni i\sqrt{3} ga qo'shish.

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

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

$$x=\frac{-\sqrt{3}i-1}{2}$$

x=\frac{1±\sqrt{3}i}{-2} tenglamasini yeching, bunda ± manfiy. 1 dan i\sqrt{3} ni ayirish.

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

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

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

Tenglama yechildi.

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

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.

$$-x^{2}-x-1=0$$

1 ni tenglamaning ikkala tarafiga qo'shish.

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

O‘zidan -1 ayirilsa 0 qoladi.

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

0 dan -1 ni ayirish.

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

Ikki tarafini -1 ga bo‘ling.

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

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

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

-1 ni -1 ga bo'lish.

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

1 ni -1 ga bo'lish.

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

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.

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

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

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

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

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

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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

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

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

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