This week has been a grind. Parabolas have probably been the toughest subject for me to understand in the 17 weeks I’ve been working at this. As I write this, I wouldn’t have thought quadratic equations would be that much more difficult than linear equations but at this point I find them to be much, much tougher. AGH!! That said, I just finished the unit test and got 100% on the first try! However, as I was going through it, I was the least sure of my answers on this test than of any other test so far. As I’ve mentioned before, there are times when I come back to a subject and find it much easier the second time through and I hope (and am optimistically confident) that’s what’s going to happen with quadratics.

This week, right off the bat I was taught the Quadratic Formula. As I mentioned at the end of my last post, I don’t know how to write the radical symbol on here but here is the formula as best as I can write it:

• Quadratic written in Standard Form:
  • ax^2 + bx + c = 0
• Quadratic Formula
  • (the brackets { and } equal square root)
  • x = (-b +/- {b^2 -4ac})/2a

This formula is used to find the “roots” or the “zeros” of a parabola. In my mind I think of the roots/zeros as the x-intercept since they are the points where the parabola crosses the x-axis. I’ve never heard to it referred to as the x-intercept, however, so I’m sure there’s a reason why they’re not the same thing. One thing I sometimes forget to do (which is partly why I’m writing it down here in hopes that I will remember in the future!) is to factor out any perfect squares in the square root. For example:

• if x = (-b +/- {44})2/a, then
• x = (-b +/- 2{11})/2a

In the Q.F., the section b^2 -4ac is referred to as the Discriminate. This part of the equation can quickly tell you if the parabola coming crosses over the x-axis, a.k.a. if it has solutions.

• If b^2 -4ac is > 0, there are two solutions (i.e. it crosses the x-axis in two places),
• if b^2 -4ac is = 0, there is one solution (i.e. the vertex of the parabola is on the x-axis), and
• if b^2 -4ac is < 0 there are no solutions (i.e., you guessed it, it doesn’t cross the x-axis).

Getting from standard form to the quadratic equation is about a 9-10 step process and, for me, still somewhat difficult. Part of getting to the Q.F. from standard form is what’s called “completing the square”. Completing the square also enables you to go from Standard Form to Vertex Form and lets you find the vertex of a parabola more easily. The formula for completing the square is:

• x^2 + 2ax = 0
• x^2 + 2ax/2 + 2ax/2 + (2a/2)^2 – (2a/2)^2 = 0
•  x(x + 2a) + 2a(x + 2a) – (2a/2)^2 = 0
• (x + 2a)^2 – (2a/2)^2 = 0

Another equation you can use to find the x value of the vertex in a parabola when it’s written in standard form is x = -b/2a. Then, taking the x-value from this equation and substituting it into the initial equation written in standard from, you will get the y-value of the vertex. I don’t know how to get to or why the equation x = -b/2a works which I find frustrating when using it…

I learned that the word “concavity” used in quadratic equations refers to whether a parabola opens upwards or downwards.

Finally, the following are the equations for the different commonly used forms of quadratics and their different pros/cons:

• Standard form
  • ax^2 + bx + c
    • Easy to determine the Y-intercept
• Vertex Form (i.e. the perfect square/completing the square)
  • (x + a/2)^2 + b
    • Easy to determine the vertex
• Factored Form
  • 0 = (x + a)(x + b)
    • Easy to find the roots/zeros

I have to admit I am somewhat discouraged as I sit here writing this. I feel like I really don’t have a strong understanding of how all these quadratic formulas work. My goal is to get through Algebra 1 in the next two days in order to finish the course by the end of the year, but I feel like I’m rushing through it so fast that I’m not giving myself enough time/practice to fully comprehend the things I’m being taught. Again, I truly think that when I come back to quadratics I’ll find them easier to understand.

As bummed as I am about not having a strong understanding of quadratics, I’m very pleased that I should be able to hit my goal and get through the course in the next two days. All I need to do is get through the final unit, Irrational Numbers, and finish the course challenge. The unit Irrational Numbers doesn’t have any practice modules, however, and is only nine videos which means it will just be a matter of watching the videos and taking notes. I should be able to finish that today (Monday) and have all of tomorrow to work through the course challenge. The Leafs game starts at 6pm tomorrow so obviously I’ll need to have it finished before then. Obviously.