Equations with Complex Numbers - Algebra II
Card 0 of 7
Solve for 
:
Solve for
In order to solve this problem, we need to first simplify our equation. The first thing we should do is distribute the square, which gives us
Now 
is actually just 
. Therefore, this becomes
Now all we need to do is solve for 
in the equation:
which gives us
Finally, we get
and therefore, our solution is
In order to solve this problem, we need to first simplify our equation. The first thing we should do is distribute the square, which gives us
Now
Now all we need to do is solve for
which gives us
Finally, we get
and therefore, our solution is
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are real numbers.
Evaluate 
.
Evaluate
For two imaginary numbers to be equal to each other, their imaginary parts must be equal. Therefore, we set, and solve for 
in:
For two imaginary numbers to be equal to each other, their imaginary parts must be equal. Therefore, we set, and solve for
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Solve for 
and 
:
Solve for
Remember that
So the powers of 
are cyclic.This means that when we try to figure out the value of an exponent of 
, we can ignore all the powers that are multiples of 
because they end up multiplying the end result by 
, and therefore do nothing.
This means that
Now, remembering the relationships of the exponents of 
, we can simplify this to:
Because the elements on the left and right have to correspond (no mixing and matching!), we get the relationships:
No matter how you solve it, you get the values 
, 
.
Remember that
So the powers of
This means that
Now, remembering the relationships of the exponents of
Because the elements on the left and right have to correspond (no mixing and matching!), we get the relationships:
No matter how you solve it, you get the values
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If 
and 
are real numbers, and 
, what is 
if 
?
If
To solve for 
, we must first solve the equation with the complex number for 
and 
. We therefore need to match up the real portion of the compex number with the real portions of the expression, and the imaginary portion of the complex number with the imaginary portion of the expression. We therefore obtain:
and 
We can use substitution by noticing the first equation can be rewritten as 
and substituting it into the second equation. We can therefore solve for 
:
With this 
value, we can solve for 
:
Since we now have 
and 
, we can solve for 
:
Our final answer is therefore 
To solve for
We can use substitution by noticing the first equation can be rewritten as
With this
Since we now have
Our final answer is therefore
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Solve for 
if 
.
Solve for
Go about this problem just like any other algebra problem by following your order of operations. We will first evaluate what is inside the parentheses: 
. At this point, we need to know the properties of 
which are as follows:
Therefore, 
and the original expression becomes 
Go about this problem just like any other algebra problem by following your order of operations. We will first evaluate what is inside the parentheses:
Therefore,
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Evaluate and simplify 
.
Evaluate and simplify
The first step is to evaluate the expression. By FOILing the expression, we get:
Now we need to simplify any terms that we can by using the properties of 
Therefore, the expression becomes
The first step is to evaluate the expression. By FOILing the expression, we get:
Now we need to simplify any terms that we can by using the properties of
Therefore, the expression becomes
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Solve
Solve
To solve
Subtract 
from both side:
Which is never true, so there is no solution.
To solve
Subtract
Which is never true, so there is no solution.
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