Today's soundtrack is Christmas With the Mills Family, a Spotify playlist that includes my favourite Christmas songs.
This afternoon, I'm learning how to complete the square to convert a quadratic function from the general form to the standard form. The general form is the least useful quadratic form; in it, "most characteristics of the graph cannot be identified" (Pearson, Pre-Calculus 11, p. 292), so we'll usually want to convert a quadratic equation from the general form whenever we need to work with it.
There are two different methods that we will use, depending on whether ax is positive or negative.
We use the following method to convert a quadratic function from general form to standard form if a > 0:
Remove the common factor from the first two terms by placing it outside a bracket wrapped around the first two terms. Example: y = 2x² + 16x + 24 becomes y = 2(x² + 8x) + 24
Divide the new second term by 2, then square it. Inside our brackets, add and then subtract that new number. Example: With y = 2(x² + 8x) + 24, 8/2=4; 4²=16. So our equation now becomes y = 2(x² + 8x + 16 - 16) + 24
Inside the brackets, we write the first three terms as a perfect square. Next, we remove the fourth term from the brackets, put it inside its own little set of brackets, and multiply it by 2. Example: y = 2(x² + 8x + 16 - 16) + 24 becomes y = 2(x² + 8x + 16) - 2(16) + 24
Now, we solve the equation! Example: y = 2(x² + 8x + 16) - 2(16) + 24, y = 2(x² + 8x + 16) - 32 + 24, y = 2(x² + 8x + 16) - 8. So the standard form of y = 2x² + 16x + 24 is y = 2(x² + 8x + 16) - 8. (Examples from Pearson's Pre-Calculus 11, p. 293)
I'm out of time for today; next time, I'll be learning how to convert a quadratic function from general form to standard form if a < 0.