Step-by-step Solution

Solve the quadratic equation $x^2-x+1=0$

Go!
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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 $2$

$x=0.5+0.866025i,\:x=\frac{1-1.732051i}{2}$
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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

$x=0.5+0.866025i,\:x=0.5-0.866025i$
$x^2-x+1=0$

Main topic:

Quadratic equations

Related Formulas:

1. See formulas

Time to solve it:

~ 0.05 s