Graphing complex numbers - GMAT Quantitative
Card 0 of 56
Give the 
-intercept(s) of the parabola with equation 
. Round to the nearest tenth, if applicable.
Give the
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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
In which quadrant does the complex number 
lie?
In which quadrant does the complex number
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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
In which quadrant does the complex number 
lie?
In which quadrant does the complex number
Tap to see back →
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
In which quadrant does the complex number 
lie?
In which quadrant does the complex number
Tap to see back →
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
In which quadrant does the complex number 
lie?
In which quadrant does the complex number
Tap to see back →
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
Raise 
to the power of four.
Raise
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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:
Raise 
to the power of eight.
Raise
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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:
Give the -intercept(s) of the parabola with equation . Round to the nearest tenth, if applicable.
Give the
Tap to see back →
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
In which quadrant does the complex number lie?
In which quadrant does the complex number
Tap to see back →
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
In which quadrant does the complex number lie?
In which quadrant does the complex number
Tap to see back →
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
In which quadrant does the complex number lie?
In which quadrant does the complex number
Tap to see back →
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
In which quadrant does the complex number lie?
In which quadrant does the complex number
Tap to see back →
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
Raise to the power of four.
Raise
Tap to see back →
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:
Raise to the power of eight.
Raise
Tap to see back →
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:
Give the -intercept(s) of the parabola with equation . Round to the nearest tenth, if applicable.
Give the
Tap to see back →
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
In which quadrant does the complex number lie?
In which quadrant does the complex number
Tap to see back →
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
In which quadrant does the complex number lie?
In which quadrant does the complex number
Tap to see back →
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
In which quadrant does the complex number lie?
In which quadrant does the complex number
Tap to see back →
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
In which quadrant does the complex number lie?
In which quadrant does the complex number
Tap to see back →
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
Raise to the power of four.
Raise
Tap to see back →
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: