Author: Sybil Shaver

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.

MAT1275CompletingSquare

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.

 

Today is National Voter Registration Day.

If you are eligible to vote and have not already done so, please register and vote in the upcoming  November elections. (There are many positions on the ballot, even though it’s not a Presidential election year!)

If you are already registered, you can encourage your friends and family to register as well.

In New York state, the deadline to register for this election is the 11th of October. You can find information about how to check eligibility and how to register either online, by mail, or in person, at this link.

Here are two useful tools from vote.org

iFrameResize({ log:true, checkOrigin:false});

iFrameResize({ log:true, checkOrigin:false});

 

And here, for your enjoyment: Luis Miranda Jr (father of Lin-Manuel) teaches us how to conjugate the verb “votar”

 

Note: the math notation in this page may take a minute or so to load. Please wait until the dollar signs go away and the math appears!

How the Quadratic Formula is derived:

We start with any quadratic equation in the form
$ax^{2} + bx + c = 0$
where $a$, $b$, and $c$ are numbers and $a \neq 0$^[1]

We solve this general equation for $x$ by completing the square and using the Square Root Property, which says:

If $Z^{2} = Q$, then $Z = \pm \sqrt{Q}$

OK, here we go: to solve $ax^{2} + bx + c = 0$
move the constant term $c$ to the other side:
$ax^{2} + bx = -c$
Now divide both sides by $a$ so that the leading coefficient will be 1.
$\frac{ax^{2}}{a} + \frac{bx}{a} = \frac{-c}{a}$
Simplify:
$x^{2} + \frac{b}{a}x= -\frac{c}{a}$

Now we complete the square on the left-hand side. To find what needs to be added to complete the square, compare:
$x^{2} + 2Ax + A^{2} = (x+A)^{2}$
$x^{2} + \frac{b}{a}x  +  ?$
This tells us that the coefficient of $x$, which is $2A$ in the first line, must equal the coefficient of $x$ in the second line, which is $\frac{b}{a}$
So  $2A = \frac{b}{a}$
$\implies A = \frac{1}{2}\left(\frac{b}{a}\right) = \frac{b}{2a}$^[2]
The number we have to add to complete the square is
$A^{2} =  \left(\frac{b}{2a}\right)^{2} = \frac{b^{2}}{4a^{2}}$

Add this to both sides of the equation and we get:
$x^{2} + \frac{b}{a}x + \frac{b^{2}}{4a^{2}} = -\frac{c}{a} + \frac{b^{2}}{4a^{2}}$
We’ll simplify the right-hand side by combining over a common denominator  $4a^{2}$^[3].
To change $-\frac{c}{a}$ so that it has the common denominator, we need to multiply the top and bottom by $4a$:
$-\frac{c}{a}\cdot \frac{4a}{4a} = -\frac{4ac}{4a^{2}}$
So the right-hand side of the equation becomes
$-\frac{c}{a} + \frac{b^{2}}{4a^{2}} = -\frac{4ac}{4a^{2}} + \frac{b^{2}}{4a^{2}} = \frac{-4ac + b^{2}}{4a^{2}} = \frac{b^{2} – 4ac}{4a^{2}}$

Now our equation looks like this:
$x^{2} + \frac{b}{a}x + \frac{b^{2}}{4a^{2}} = \frac{b^{2} – 4ac}{4a^{2}}$
Write the left-hand side in its factored form so we can use the Square Root Property.
$\left(x + \frac{b}{2a}\right)^{2} =  \frac{b^{2} – 4ac}{4a^{2}}$
The Square Root Property tells us that
If  $\left(x + \frac{b}{2a}\right)^{2} =  \frac{b^{2} – 4ac}{4a^{2}}$
then  $x + \frac{b}{2a} = \pm \sqrt{ \frac{b^{2} – 4ac}{4a^{2}}}$

That square root on the right-hand side can be simplified:
$\sqrt{ \frac{b^{2} – 4ac}{4a^{2}}} = \frac{\sqrt{b^{2} – 4ac}}{\sqrt{4a^{2}}} = \frac{\sqrt{b^{2} – 4ac}}{2a}$ provided that $a > 0$. We’ll return to that last bit later.

So now our equation looks like
$x + \frac{b}{2a} = \pm\frac{\sqrt{b^{2} – 4ac}}{2a}$
We’re almost finished. We just have to subtract  $ \frac{b}{2a}$ from both sides and simplify a bit.
$x = -\frac{b}{2a} \pm\frac{\sqrt{b^{2} – 4ac}}{2a}$

$x = \frac{-b \pm\sqrt{b^{2} – 4ac}}{2a}$

 

[1] Why do we require that $a \neq 0$ at the beginning?

[2] Dividing by $2$ is the same as multiplying by $\frac{1}{2}$, which is easier to do since we are working with an algebraic fraction here.

[3] We have not yet worked with algebraic fractions (rational expressions) in this course, but it is coming up, so just realize that it is basically the same as working with regular numerical fractions, and do the best you can with this for now.

Test 1 is scheduled for Monday 23 September. It will be for the whole period.

 

Please make sure that you are familiar with my course policies as they apply to Tests. Make sure that you also read the CUNY Academic Integrity Policy which is on that linked page!

You are responsible for reviewing for the Tests. I have made an assignment in WeBWorK that is named Test1Review.  Use them to practice working problems. The best way to review for any test is to work the problems as if you were taking a test: don’t look at notes (or hints) and try not to take more than about 3-5 minutes per problem – except for solving a 3 by 3 system, which takes longer! But not more than about 10 minutes. Aim for that when you practice.

The WeBWorK review assignment has much more problems than will be on the Test! You will see that there is some repetition.

The most important thing to do to prepare for any Test in Math is to work problems. Practice, practice practice.

It’s also a good idea to go back to the quizzes. I’ll try to make sure you have all of them returned to you tomorrow. I have posted copies of the Quizzes which you can use to test yourself (allow 10 minutes for each Quiz) and then check against the solutions which I have also posted there.

Another thing you can do is to return to the old WeBWorK problem sets. I have enabled “Show me another” so that you can practice more.

Concentrate on the problems which give you the most trouble. It is also good to work with a classmate or a group of classmates.

You may wish to watch or re-watch the video resources we have made available. Here are some links:

Notes for students

– if you feel that you are working hard and not getting the results you want. This also includes advice on the subject of how to get the most from the video resources linked below.

How to use the video resources

MAT 1275CO syllabus with video resource links

 

And last but not least, Tutoring and other resources

 

 

 

 

This is the schedule for the Math specialist tutors who are hired under a special grant. It is separate from the tutoring in the Atrium Learning Center.

 

Perkins Fall 2019

New topics: finding the greatest common factor (GCF) and factoring it out; Factoring by grouping.

 

Vocabulary:

factor (noun)

factor (verb)

common factor

greatest common factor

term

coefficient

polynomial

 

Math specialists will be available to tutor tomorrow, Friday (and Fridays from now on) – more days will be announced soon.

Fridays beginning tomorrow from

10 am to 6 pm in MAT 065 through MAT 2675.

Tutoring will take place in room N723.

Questions for the Lines Lab:

Give a definition of x-intercept and y-intercept.

What is the y-coordinate of the x-intercept?
What is the x-coordinate of the y-intercept?

How can you use the two intercepts to find the equation of the line?

Does a line always have an x-intercept? (If not, give an example.)

Does a line always have a y-intercept? Why or why not? (If not, give an example.)

Are there any lines that have neither an x-intercept nor a y-intercept?

Here is the link for information about accessing your City Tech email account.

Notes from last time: **Still being updated!**

I solved a linear system by graphing, and another system by the method of substitution and then by the method of elimination (in two ways). You must know these methods very thoroughly and be able to use them both.

 

• Active learning practice solving a system of 2 linear equations in 2 variables by substitution and elimination

• Solving a system of 3 linear equations in 3 variables (active learning with outline notes)

The two outline notes pages which I handed out in class are also available here:

MAT1275systemsReviewClasswork

MAT12753by3systemsClasswork

There is a very nice video explaining this method at Patrick’s Just Math Tutorials. He explains the reasoning along the way.

There is a longer video explanation of the method along with background information about 3 by 3 systems at Khan Academy.

 

Make sure to answer (in your notebook) the Questions that go with the Lines Lab assignment.