Wednesday 22 March class

Posted on March 22, 2017 by Sybil Shaver

Topics:

• More on using the Square Root Property (from discussion of homework problems)

• Perfect squares of binomials (perfect square trinomials) and Completing the Square

• Quick look at solving quadratic equations by completing the square, to get to…

• Derivation of the Quadratic Formula

• Using the Quadratic Formula to solve a quadratic equation (more next time on this)

 

I am working on notes on the derivation of the Quadratic Formula and will post them here as soon as they are done.

Completing the square is not a method that is commonly used to solve a quadratic equation (and we will not really use it for this purpose): its main role here is to allow us to show where the Quadratic Formula comes from. (The Quadratic Formula is actually a theorem of mathematics, and the derivation of the formula is the proof of the theorem!)

There is a questionable moment in the derivation, when we take the square root of 4a^{2}, and the textbook (and most sources) just says that \sqrt{4a^{2}} = 2a without further comment. But that is not necessarily true! It is not necessarily true, because a could possibly be a negative number: we know that \sqrt{a^{2}} = |a|. It turns out that you get the same end result even in the case where a is negative, but we should not pass over that without comment. My notes will show how it works out.

It is always important to remember that \sqrt{x^{2}} = x only if x \ge 0! Otherwise you can get into a lot of trouble. We made this point when discussing the Square Root Property. There are a lot of sources that write it incorrectly, and even people who should know better do that. (I’m not going to write the incorrect version here, because I don’t want to put it into your heads.) Just remember that

For any real number NUMBER

If  \left(\text{THING}\right)^{2} = NUMBER,

then  \text{THING} = \sqrt{NUMBER}  or  \text{THING} = -\sqrt{NUMBER}

Or, in shorthand,

\text{THING} = \pm\sqrt{NUMBER}

Comment on using the Quadratic Formula: The Quadratic Formula can be used to solve any quadratic equation at all: in that respect it is superior to the other two methods (factoring and using the Zero Product Principle, or using the Square Root Property) which can only be used when the equation we are trying to solve has a certain form. However, it is always better to use factoring or the square root property if possible! So you should examine your equation to see if one of those two methods can be used before trying the Quadratic Formula (unless you are specifically instructed to use the Quadratic Formula). There are several reasons to prefer those other methods when they can be used: one reason is that human beings, as opposed to computing machines, tend to make errors when using the QF, either by not correctly identifying the numbers a, b, and c, or by not doing the operations in the correct order, or by not having the formula correct in the first place. Another reason is that it is important to practice factoring polynomials, because factoring is used for so many other things. The more practice you do, the more easily you will be able to factor when you need to do so. Another reason is that factoring or using the square root property, when it is possible to do so, will almost always be a lot faster than using the QF. The QF is very useful and important, but never forget that you have other options available!

Homework:

• Review the discussion of the homework problems. Make sure that you understand the corrections we made.

• Review completing the square. It is very important that you see how this is coming from the special product patterns

\left(x+B\right)^{2} = x^{2} +2Bx + B^{2}

and

\left(x-B\right)^{2} = x^{2} -2Bx + B^{2}

(which you should always be using when the occasion arises). It is much easier to remember how to complete the square if you know why you are doing what you are doing. And we will need to be able to complete squares for other purposes later in the course.

• Do the following from the textbook: p. 581 #21-32 all; p. 595 #5-25 odd

Whenever you use the Quadratic Formula, it is an excellent idea to actually write it out and say it out loud while you do so, before substituting numbers in. This will help fix it in your memory.

• Do the WeBWorK, which has two (short) parts.

Note: in the assignment on the Quadratic Formula, the problems are phrased in the following form: “Find the roots of the parabola y=x^{2} - 3x +2 (for example). Finding the roots (more correctly, the x-intercepts) means exactly the same thing as asking you to solve the equation x^{2} - 3x +2=0.

Remember that you can use the Piazza discussion board to ask questions if you get stuck on any of the WeBWorK or the other homework problems. Please be as specific as possible about what you are stuck on and don’t forget to put the problem itself into your question, as we may not have the textbook with us when we go to answer it!

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