Find the Roots of Complex Numbers - Pre-Calculus
Card 0 of 52
Solve for 
(there may be more than one solution).
Solve for 
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To solve for the roots, just set equal to zero and solve for z using the quadratic formula (
) : 
and now setting both 
and 
equal to zero we end up with the answers 
and 
To solve for the roots, just set equal to zero and solve for z using the quadratic formula (
Evaluate 
, where 
is a natural number and 
is the complex number 
.
Evaluate
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Note that,
Note that,
What is the length of
?
What is the length of
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We have
.
Hence,
.
We have
Hence,
Solve for 
(there may be more than one solution).
Solve for
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Solving that equation is equivalent to solving the roots of the polynomial 
.
Clearly, one of roots is 1.
Thus, we can factor the polynomial as 
so that we solve for the roots of 
.
Using the quadratic equation, we solve for roots, which are 
.
This means the solutions to 
are
Solving that equation is equivalent to solving the roots of the polynomial
Clearly, one of roots is 1.
Thus, we can factor the polynomial as
so that we solve for the roots of
Using the quadratic equation, we solve for roots, which are
This means the solutions to
Recall that 
is just shorthand for 
when dealing with complex numbers in polar form.
Express 
in polar form.
Recall that
Express in polar form.
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First we recognize that we are trying to solve 
where 
.
Then we want to convert 
into polar form using,
and 
.
Then since De Moivre's theorem states,
if 
is an integer, we can say
.
First we recognize that we are trying to solve
Then we want to convert
Then since De Moivre's theorem states,
Compute 
Compute
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To solve this question, you must first derive a few values and convert the equation into exponential form: 
: 
Now plug back into the original equation and solve: 
To solve this question, you must first derive a few values and convert the equation into exponential form: 
Now plug back into the original equation and solve:
Solve for all possible solutions to the quadratic expression: 
Solve for all possible solutions to the quadratic expression:
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Solve for complex values of m using the aforementioned quadratic formula:
Solve for complex values of m using the aforementioned quadratic formula:
Determine the length of 
Determine the length of
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, so 
Solve for 
(there may be more than one solution).
Solve for 
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To solve for the roots, just set equal to zero and solve for 
using the quadratic formula (
): 
and now setting both 
and 
equal to zero we end up with the answers 
and 
.
To solve for the roots, just set equal to zero and solve for 
Which of the following lists all possible solutions to the quadratic expression: 
Which of the following lists all possible solutions to the quadratic expression:
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Solve for complex values of 
using the quadratic formula: 
Solve for complex values of 
Solve for 
(there may be more than one solution).
Solve for 
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To solve for the roots, just set equal to zero and solve for z using the quadratic formula, which is 
and now setting both 
and 
equal to zero we end up with the answers 
and 
and so the correct answer is 
.
To solve for the roots, just set equal to zero and solve for z using the quadratic formula, which is
Solve for all possible solutions to the quadratic expression: 
Solve for all possible solutions to the quadratic expression:
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Solve for complex values of 
using the quadratic formula: 
.
Solve for complex values of 
Determine the length of 
.
Determine the length of 
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To begin, we must recall that 
. Plug this in to get 
. Length must be a positive value, so we'll take the absolute value: 
. Therefore the length is 3.
To begin, we must recall that 
Solve for (there may be more than one solution).
Solve for
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To solve for the roots, just set equal to zero and solve for z using the quadratic formula () : and now setting both and equal to zero we end up with the answers and
To solve for the roots, just set equal to zero and solve for z using the quadratic formula (
Evaluate , where is a natural number and is the complex number .
Evaluate
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Note that,
Note that,
What is the length of
?
What is the length of
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We have
.
Hence,
.
We have
Hence,
Solve for (there may be more than one solution).
Solve for
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Solving that equation is equivalent to solving the roots of the polynomial .
Clearly, one of roots is 1.
Thus, we can factor the polynomial as
so that we solve for the roots of .
Using the quadratic equation, we solve for roots, which are .
This means the solutions to are
Solving that equation is equivalent to solving the roots of the polynomial
Clearly, one of roots is 1.
Thus, we can factor the polynomial as
so that we solve for the roots of
Using the quadratic equation, we solve for roots, which are
This means the solutions to
Recall that is just shorthand for when dealing with complex numbers in polar form.
Express in polar form.
Recall that
Express in polar form.
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First we recognize that we are trying to solve where .
Then we want to convert into polar form using,
and .
Then since De Moivre's theorem states,
if is an integer, we can say
.
First we recognize that we are trying to solve
Then we want to convert
Then since De Moivre's theorem states,
Compute
Compute
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To solve this question, you must first derive a few values and convert the equation into exponential form: :
Now plug back into the original equation and solve:
To solve this question, you must first derive a few values and convert the equation into exponential form:
Now plug back into the original equation and solve:
Solve for all possible solutions to the quadratic expression:
Solve for all possible solutions to the quadratic expression:
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Solve for complex values of m using the aforementioned quadratic formula:
Solve for complex values of m using the aforementioned quadratic formula: