Final Answer
$x=0.5+0.866025i,\:x=0.5-0.866025i$
Step-by-step Solution
Problem to solve:
$x^2-x+1=0$
1
To find the roots of a polynomial of the form $ax^2+bx+c$ we use the quadratic formula, where in this case $a=1$, $b=-1$ and $c=1$. Then substitute the values of the coefficients of the equation in the quadratic formula:
- $\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
$x=\frac{1\pm \sqrt{-3}}{2}$
2
To obtain the two solutions, divide the equation in two equations, one when $\pm$ is positive ($+$), and another when $\pm$ is negative ($-$)
$x=\frac{1+\sqrt{-3}}{2},\:x=\frac{1-1\cdot \sqrt{-3}}{2}$
3
Calculate the power $\sqrt{-3}$ using complex numbers
$x=\frac{1+1.732051i}{2},\:x=\frac{1-1\cdot \sqrt{-3}}{2}$
4
Calculate the power $\sqrt{-3}$ using complex numbers
$x=\frac{1+1.732051i}{2},\:x=\frac{1-1.732051i}{2}$
5
Split the fraction $\frac{1+\frac{3}{\sqrt{3}}i}{2}$ in two fractions with common denominator $2$
$x=0.5+0.866025i,\:x=\frac{1-1.732051i}{2}$
6
Split the fraction $\frac{1-\frac{3}{\sqrt{3}}i}{2}$ in two fractions with common denominator $2$
$x=0.5+0.866025i,\:x=0.5-0.866025i$
Final Answer
$x=0.5+0.866025i,\:x=0.5-0.866025i$