Final answer to the problem
$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{3}i}{2}$
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Step-by-step Solution
How should I solve this problem?
- Choose an option
- Solve for x
- Find the derivative using the definition
- Solve by quadratic formula (general formula)
- Simplify
- Find the integral
- Find the derivative
- Factor
- Factor by completing the square
- Find the roots
- Load more...
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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: $\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
$x=\frac{- -1\pm \sqrt{{\left(-1\right)}^2-4\cdot 1}}{2}$
Simplifying
$x=\frac{- -1\pm \sqrt{{\left(-1\right)}^2-4\cdot 1}}{2}$
Any expression multiplied by $1$ is equal to itself
$x=\frac{- -1\pm \sqrt{{\left(-1\right)}^2-4}}{2}$
Multiply $-1$ times $-1$
$x=\frac{1\pm \sqrt{{\left(-1\right)}^2-4}}{2}$
Calculate the power ${\left(-1\right)}^2$
$x=\frac{1\pm \sqrt{1-4}}{2}$
Add the values $1$ and $-4$
$x=\frac{1\pm \sqrt{-3}}{2}$
Calculate the power $\sqrt{-3}$ using complex numbers
$x=\frac{1\pm \sqrt{3}i}{2}$
2
Simplifying
$x=\frac{1\pm \sqrt{3}i}{2}$
3
To obtain the two solutions, divide the equation in two equations, one when $\pm$ is positive ($+$), and another when $\pm$ is negative ($-$)
$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{3}i}{2}$
4
Combining all solutions, the $2$ solutions of the equation are
$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{3}i}{2}$
Final answer to the problem
$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{3}i}{2}$
