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

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

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Final Answer

$x=0.5+0.866025i,\:x=0.5-0.866025i$

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Step-by-step Solution

Problem to solve:

$x^2-x+1=0$

Choose the solving method

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:

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

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

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

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

Calculate the power $\sqrt{-3}$ using complex numbers

$x=\frac{1+1.732051i}{2},\:x=\frac{1-\sqrt{-3}}{2}$

Calculate the power $\sqrt{-3}$ using complex numbers

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

Expand the fraction $\frac{1+1.732051i}{2}$ into $2$ simpler fractions with common denominator $2$

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

Take $\frac{\sqrt{3}}{2}$ out of the fraction

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

Simplify

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

Expand the fraction $\frac{1-1.732051i}{2}$ into $2$ simpler fractions with common denominator $2$

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

Take $\frac{-\sqrt{3}}{2}$ out of the fraction

$x=0.5+0.866025i,\:x=0.5-0.866025i$

Simplify

$x=0.5+0.866025i,\:x=0.5-0.866025i$

Final Answer

$x=0.5+0.866025i,\:x=0.5-0.866025i$

Step-by-step Solution