The topics: solving quadratic equations using completing the square; complex numbers (basic definitions); using the quadratic formula.
Complex Numbers: basic definitions
To start with, we need to know what the Real numbers are. You can think of the Real numbers as the numbers which correspond to points on the number line. There is more information here [Purplemath].
More on the real and complex numbers from Math is Fun:
Common number types (vocabulary)
Notice that a negative real number like -1 has no square roots which are real numbers, because any real number, when squared, gives a result which is greater than or equal to 0.
We fix this problem by giving a name to a square root of -1:
Definition: $\sqrt{-1} = i$
-1 has a second square root, namely, $-i$
Using $i$ we can write the square root of any negative real number, as we will see.
Definition: An imaginary number is a number which can be put into the form $bi$, where $b$ is a real number, $b \neq 0$.
Definition: A complex number is a number which can be put into the form $a + bi$, where $a$ and $b$ are real numbers (one or both of which may be 0).
Definition: For a complex number $a + bi$,
• $a$ is called the real part
• $b$ is called the imaginary part
Definition: Two complex numbers are called conjugate if their real parts are the same and their imaginary parts are the same except that they have opposite signs.
Example: $-3+4i$ and $-3-4i$ are conjugate.
We use the same language for solutions to quadratic equations which have radicals in them: (remember that $i = \sqrt{-1}$ is essentially a radical!)
$2 – 2\sqrt{3}$ and $2+2\sqrt{3}$ are conjugate.
The importance of conjugates will be seen as we work with solving quadratic equations and with complex numbers.
Here is the handout on solving quadratic equations by completing the square and using the Square Root Property: you should complete the two examples we did not do in class.
Notice as you get your solutions, and also to the problems in the WeBWorK about the Square Root Property, what happens when you get solutions which are not integers? There are actually several distinguishable types of results. Make a list for the problems on the handout and for problems #7, 8, 9 of the SquareRootProperty WeBWorK of which fall in these categories: make sure to put the problem number and the assignment it came from, and also copy the equation you had to solve. We’ll need to look at the equations in more detail.
CATEGORIES OF SOLUTIONS OF QUADRATIC EQUATIONS:
• Do you always get two distinct solutions? If not, which problem or problems gave only one solution?
• Which problem or problems gave integer solutions?
• Which problem or problems gave rational number solutions?
• Which problem or problems gave real number irrational solutions?
• Which problem or problems gave nonreal complex number solutions?
Please keep this in your notebook: you will add to the lists as you go through the assignment on Quadratic Formula in WeBWorK.
Homework problems: There are several assignments open, so here is the recommended order and timing for them
• Work the WeBWorK assignment SquareRootProperty: it is due tomorrow night, but try to finish it tonight. Also list the problems 7-9 in the categories given above.
• Work the problems on the handout MAT1275CompletingSquare
also listing the problems in the categories given above. Do this before class tomorrow.
• Work the problems in the WeBWorK QuadraticFormula. There are only 3 problems, so try to finish them before class tomorrow. If you get a problem wrong, you will be given a problem which will lead you through the quadratic formula step-by-step. If you get the problem correct, you won’t see this, so don’t worry about it.
For these problems, also list them in the categories above.
We will take a look at those categories in a future class.
