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Showing posts with label complex. Show all posts
Showing posts with label complex. Show all posts

## Friday, November 9, 2012

### Chapter 9 Sample Test Questions

a. Solve by completing the square.
b. Solve using the quadratic formula.
a. Solve by factoring.
b. Solve using the quadratic formula.

Example: Graph. Label the x- and y-intercepts as well as the vertex.

Projectile Problem: An object is launched from an 80-foot tower with an initial speed of 75 feet per second. At what times is the object 120 feet high? Use the following function:
Geometry Problem: The height of a triangle is 2 centimeters more than 3 times the base. If the area is 28 square centimeters then what is the length of the base and height?

Example: Simplify
a.

b.

### Complex Numbers and Complex Solutions

Up to this point, we have been working exclusively with real numbers.  Now we shed this limitation and allow for a much broader range of problems.
With these definitions, we have greatly expanded our space of numbers.  Notice that any real number is also a complex number, for example 5 = 5 + 0i.  Here the real part is 5 and the imaginary part is 0.  Next we consider powers of i.

Instructional Video: Introduction to Complex Numbers

Simplify.
Adding and subtracting complex numbers is just a matter of adding like terms. Be sure to use the order of operations and add real and imaginary parts separately.

We have to use a bit of caution when multiplying complex numbers.  First, we will run into i^2 often. In this case, we will replace them all with -1.  In addition, the property
is true only when either A or B is non-negative; so it is best to simplify using the imaginary unit before multiplying.
Tip: Make use of the imaginary unit if the radicand is negative before trying to simplify.

Multiply.
Multiplying complex numbers often requires the distributive property.
Dividing complex numbers requires techniques similar to rationalizing the denominator.

Divide.
When a complex number is in the denominator, multiply numerator and denominator by its conjugate.
Sometimes quadratic equations have complex solutions.

Solve by extracting roots.
Solve by completing the square.