Complete the Square

mathdad

mathdad

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To complete the square of x^2 + bx, we add (b/2)^2, which is (b^2)/4. Do we add (b/2)^2 on both sides of a given quadratic equation? I say yes. You? When is it necessary to complete the square?
 
In order for an equation to maintain its equality, you must do the same thing to both sides. It is necessary to complete the square to solve a quadratic equation when it doesn't have rational roots and can be factored, and you're not going to use the quadratic formula.
 
MarkFL said:
In order for an equation to maintain its equality, you must do the same thing to both sides. It is necessary to complete the square to solve a quadratic equation when it doesn't have rational roots and can be factored, and you're not going to use the quadratic formula.
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I think the quadratic formula is far easier than completing the square. This is my opinion.
 
The Quadratic Formula IS Completing the Square. Try Completing the Square on ax^2 + bx + c = 0. You will gain greater understanding of the non-magical nature of the Quadratic Formula.

You must use ALL the tools in your box. Don't throw out what you might not need today.
 
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tkhunny said:
The Quadratic Formula IS Completing the Square. Try Completing the Square on ax^2 + bx + c = 0. You will gain greater understanding of the non-magical nature of the Quadratic Formula.

You must use ALL the tools in your box. Don't throw out what you might not need today.
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It is certainly worth tackling this general equation.
 
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mathdad said:
I think the quadratic formula is far easier than completing the square. This is my opinion.
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There are a couple reasons for knowing how to complete the square.

One is that although in one sense the quadratic formula is easier, it is also riskier, because unlike algebraic procedures, there is nothing to tell you if you make a mistake. You just plug in numbers, evaluate, and hope you did it right. When you carry out the whole procedure, you can check what you do at each step. With the formula, all you can really do is to put your solutions back into the original equation (which most students are too lazy to do when it involves irrational or complex numbers) to see if it works.

Perhaps more important in the long run, there are several other situations besides solving quadratic equations where completing the square is useful, and perhaps has no alternative. These include graphing general conic sections; integrating certain forms containing quadratic polynomials in calculus; and so on.
 
Dr.Peterson said:
There are a couple reasons for knowing how to complete the square.

One is that although in one sense the quadratic formula is easier, it is also riskier, because unlike algebraic procedures, there is nothing to tell you if you make a mistake. You just plug in numbers, evaluate, and hope you did it right. When you carry out the whole procedure, you can check what you do at each step. With the formula, all you can really do is to put your solutions back into the original equation (which most students are too lazy to do when it involves irrational or complex numbers) to see if it works.

Perhaps more important in the long run, there are several other situations besides solving quadratic equations where completing the square is useful, and perhaps has no alternative. These include graphing general conic sections; integrating certain forms containing quadratic polynomials in calculus; and so on.
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Thank you for clearing up my confusion. A few questions for you below (not a math question(s)).

1. Are you a math professor or retired math professor?

2. Why did you decide to volunteer your time online helping people with math?

3. What is the "best" way to get better at solving word problems?

4. Due to limited time (working 40 overnight hours per week), I decided to answer no more than 20 questions per chapter sections in the Michael Sullivan College Algebra 9th Edition textbook. I MUST get 14/20 correct answers to proceed in the book. What do you suggest in terms of study time for a person who is not a student and works full-time?
 
5. Are you just generally awesome? Yes. I am not a site spokesman, but it is easy enough to figure out for yourself.
 
mathdad said:
Thank you for clearing up my confusion. A few questions for you below (not a math question(s)).

1. Are you a math professor or retired math professor?

2. Why did you decide to volunteer your time online helping people with math?

3. What is the "best" way to get better at solving word problems?

4. Due to limited time (working 40 overnight hours per week), I decided to an It is hoped taht you try othersswer no more than 20 questions per chapter sections in the Michael Sullivan College Algebra 9th Edition textbook. I MUST get 14/20 correct answers to proceed in the book. What do you suggest in terms of study time for a person who is not a student and works full-time?
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What do you do if you do not achieve 14/20? It is hoped that you try others, not just the same 20. Can you REALLY say that you have achieved sufficient mastery with 14/20?
 
With the exception of certain specialized problems (see Dr. P's post), I recommend that those students whom I tutor use the quadratic formula if they do not quickly see a convenient factoring of the quadratic expression. My point in that recommendation is that students are typically evaluated using timed tests so that they need to practice solving problems using a speedy technique.

I might quibble with the proposition that the quadratic formula is completing the square. The quadratic formula is derived from and proved by completion of the square. The formula is a short cut for getting the result that completing the square gets. However, as Dr. P says, it can be tedious to check your work, which students tend to skip in any case.
 
mathdad said:
Thank you for clearing up my confusion. A few questions for you below (not a math question(s)).

1. Are you a math professor or retired math professor?

2. Why did you decide to volunteer your time online helping people with math?

3. What is the "best" way to get better at solving word problems?

4. Due to limited time (working 40 overnight hours per week), I decided to answer no more than 20 questions per chapter sections in the Michael Sullivan College Algebra 9th Edition textbook. I MUST get 14/20 correct answers to proceed in the book. What do you suggest in terms of study time for a person who is not a student and works full-time?
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1,2: As for me personally, I started volunteering online in 1998 while I was a software engineer, which reminded me how much I enjoyed teaching in grad school; so when my job went away, I switched to teaching math (from which I am now gradually retiring). I did all three things for the same reason: the pleasure of seeing something "click".

3: How to get better at word problems? Same answer as "How can I get to Carnegie Hall?": practice. But you know that.

4: I loved what one student told me: "I keep working at a problem until I can't not get it right." (She was an adult student, sitting in the front row, and did what she said.) That means not just counting the number of errors, but seeking deeper understanding, so that you know why you do each step, and it would feel wrong to do something invalid. For word problems in particular, that means clearly defining variables, writing equations that define the problem, and not trying to solve anything until you are sure your equations are correct.

It sounds like you have one great advantage over students: You can take the time you need to do this "mastery learning", because no one outside yourself is setting your schedule. Total amount of time is not the main issue; but probably it would be good to do your studying in large enough "chunks" that you can get deeply into a series of related problems without having to divert your attention, and can therefore apply what you learn from one problem to the next, setting it firmly into your mind.
 
tkhunny said:
What do you do if you do not achieve 14/20? It is hoped that you try others, not just the same 20. Can you REALLY say that you have achieved sufficient mastery with 14/20?
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If I cannot reach 14/20, I go back to the start of that particular section to try again and again, if needed.
Dr.Peterson said:
1,2: As for me personally, I started volunteering online in 1998 while I was a software engineer, which reminded me how much I enjoyed teaching in grad school; so when my job went away, I switched to teaching math (from which I am now gradually retiring). I did all three things for the same reason: the pleasure of seeing something "click".

3: How to get better at word problems? Same answer as "How can I get to Carnegie Hall?": practice. But you know that.

4: I loved what one student told me: "I keep working at a problem until I can't not get it right." (She was an adult student, sitting in the front row, and did what she said.) That means not just counting the number of errors, but seeking deeper understanding, so that you know why you do each step, and it would feel wrong to do something invalid. For word problems in particular, that means clearly defining variables, writing equations that define the problem, and not trying to solve anything until you are sure your equations are correct.

It sounds like you have one great advantage over students: You can take the time you need to do this "mastery learning", because no one outside yourself is setting your schedule. Total amount of time is not the main issue; but probably it would be good to do your studying in large enough "chunks" that you can get deeply into a series of related problems without having to divert your attention, and can therefore apply what you learn from one problem to the next, setting it firmly into your mind.
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Thank you for your answers. I am about 11 years away from retiring. I have been working since my early twenties in jobs without a pension or the famous 401k. I got hired last year (2018) at a worldwide famous NYC museum to work overnight hours. I will not give the museum name because you never know who is searching the internet.

Back in my college days, I fell in love with math and anxiously wanted to teach for a living. However, I was too deep in my sociology courses to switch from that career path to math. I was about two semesters away from graduation not having taken calculus 1. Switching majors meant more years on the campus. By this time in my life, I was already seriously tired of school.
 
JeffM said:
With the exception of certain specialized problems (see Dr. P's post), I recommend that those students whom I tutor use the quadratic formula if they do not quickly see a convenient factoring of the quadratic expression. My point in that recommendation is that students are typically evaluated using timed tests so that they need to practice solving problems using a speedy technique.

I might quibble with the proposition that the quadratic formula is completing the square. The quadratic formula is derived from and proved by completion of the square. The formula is a short cut for getting the result that completing the square gets. However, as Dr. P says, it can be tedious to check your work, which students tend to skip in any case.
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I enjoy the quadratic formula. I am not lazy like most young people today. I actually do check my answers by plugging the roots back into the original quadratic equations.
 
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