Solve the equation x+1+x^2=0

x^2+x+1=0

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Answer

$x_1=-0.5+0.866i,\:x_2=-0.5-0.866i$

Step by step solution

Problem

$x^2+x+1=0$
1

To find the roots of a polynomial of the form $ax^2+bx+c$ we use the quadratic formula, where $a=1$, $b=1$ and $c=1$

$x =\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
2

Substituting the values of the coefficients of the equation in the quadratic formula

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

Calculate the power

$x=\frac{-1\pm \sqrt{1-4}}{2}$
4

Add the values $1$ and $-4$

$x=\frac{-1\pm \sqrt{-3}}{2}$
5

Calculate the power using complex numbers

$x=\frac{-1\pm \sqrt{2}i}{2}$
6

To obtain the two solutions, divide the equation in two equations, one when $\pm$ is positive ($+$), and another when $\pm$ is negative ($-$)

$x_1=\frac{-1+ \sqrt{2}i}{2}\:\:,\:\:x_2=\frac{-1- \sqrt{2}i}{2}$
7

Simplifying

$x_1=-0.5+0.866i,\:x_2=-0.5-0.866i$
8

We found that the two complex solutions of the equation are

$x_1=-0.5+0.866i,\:x_2=-0.5-0.866i$

Answer

$x_1=-0.5+0.866i,\:x_2=-0.5-0.866i$

Problem Analysis

Main topic:

Quadratic formula

Time to solve it:

0.27 seconds

Views:

206