Graphing complex numbers - GMAT Quantitative Reasoning
Card 0 of 7
Give the 
-intercept(s) of the parabola with equation 
. Round to the nearest tenth, if applicable.
Give the
The 
-coordinate(s) of the 
-intercept(s) are the real solution(s) to the equation 
. We can use the quadratic formula to find any solutions, setting 
- the coefficients of the expression.
An examination of the discriminant 
, however, proves this unnecessary.
The discriminant being negative, there are no real solutions, so the parabola has no 
-intercepts.
The
An examination of the discriminant
The discriminant being negative, there are no real solutions, so the parabola has no
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In which quadrant does the complex number 
lie?
In which quadrant does the complex number
When plotting a complex number, we use a set of real-imaginary axes in which the x-axis is represented by the real component of the complex number, and the y-axis is represented by the imaginary component of the complex number. The real component is 
and the imaginary component is 
, so this is the equivalent of plotting the point 
on a set of Cartesian axes. Plotting the complex number on a set of real-imaginary axes, we move 
to the left in the x-direction and 
up in the y-direction, which puts us in the second quadrant, or in terms of Roman numerals:
When plotting a complex number, we use a set of real-imaginary axes in which the x-axis is represented by the real component of the complex number, and the y-axis is represented by the imaginary component of the complex number. The real component is
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In which quadrant does the complex number 
lie?
In which quadrant does the complex number
If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:
We are essentially doing the same as plotting the point 
on a set of Cartesian axes. We move 
units right in the x direction, and 
units down in the y direction, which puts us in the fourth quadrant, or in terms of Roman numerals:
If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:
We are essentially doing the same as plotting the point
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In which quadrant does the complex number 
lie?
In which quadrant does the complex number
If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:
We are essentially doing the same as plotting the point 
on a set of Cartesian axes. We move 
units left of the origin in the x direction, and 
units down from the origin in the y direction, which puts us in the third quadrant, or in terms of Roman numerals:
If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:
We are essentially doing the same as plotting the point
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In which quadrant does the complex number 
lie?
In which quadrant does the complex number
If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:
We are essentially doing the same as plotting the point 
on a set of Cartesian axes. We move 
units right of the origin in the x direction, and 
units up from the origin in the y direction, which puts us in the first quadrant, or in terms of Roman numerals:
If we graphed the given complex number on a set of real-imaginary axes, we would plot the real value of the complex number as the x coordinate, and the imaginary value of the complex number as the y coordinate. Because the given complex number is as follows:
We are essentially doing the same as plotting the point
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Raise 
to the power of four.
Raise
Squaring an expression, then squaring the result, amounts to taking the original expression to the fourth power. Therefore, we can first square 
:
Now square this result:
Squaring an expression, then squaring the result, amounts to taking the original expression to the fourth power. Therefore, we can first square
Now square this result:
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Raise 
to the power of eight.
Raise
For any expression 
, 
. That is, we can raise an expression to the power of eight by squaring it, then squaring the result, then squaring that result.
First, we square:
Square this result to obtain the fourth power:
Square this result to obtain the eighth power:
For any expression
First, we square:
Square this result to obtain the fourth power:
Square this result to obtain the eighth power:
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