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From Wikipedia

How to Solve It

How to Solve It (1945) is a small volume by mathematician George Pólya describing methods of problem solving.

Four principles

How to Solve It suggests the following steps when solving a mathematical problem:

  1. First, you have to understand the problem.
  2. After understanding, then make a plan.
  3. Carry out the plan.
  4. Look backon your work.How could it be better?

If this technique fails, Pólya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it." Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"

First principle: Understand the problem

"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:

  • What are you asked to find or show?
  • Can you restate the problem in your own words?
  • Can you think of a picture or a diagram that might help you understand the problem?
  • Is there enough information to enable you to find a solution?
  • Do you understand all the words used in stating the problem?
  • Do you need to ask a question to get the answer?

The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.

Second principle: Devise a plan

Pólya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:

  • Guess and check
  • Make an orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Use direct reasoning
  • Solve an equation

Also suggested:

  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward
  • Use a formula
  • Be creative
  • Use your head/noggin

Third principle: Carry out the plan

This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled; this is how mathematics is done, even by professionals.

Fourth principle: Review/extend

Pólya mentions that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.

The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:

The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.

The book has achieved "classic" status because of its considerable influence (see the next section).

Other books on problem solving are often related to more creative and less concrete techniques. See lateral thinking, mind mapping, brainstorming, and creative problem solving.

Influence

  • It has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.
  • Marvin Minsky said in his influential paper Steps Toward Artificial Intelligence that "everyone should know the work of George Pólya on how to solve problems."
  • Pólya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.
  • Russian physicistZhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with Pólya's famous book.
  • Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels Pólya's work.


From Yahoo Answers

Question:1.x= log^7(30) 2.f^(x^2-3) = 71 How do you do them right? Heres what i did, and i was wrong log7+log30 .845 + 1.47= 2.3222 and for the second i did log 72 divided by log 5 = xsquared - 3 which was 2.6572 = x squared - 3 then add 3 to get 5.6571 = x squared then do the square root of 5.65.. which is 2.3784 but these are wrong can anyone tell me why? Q2 is 5^(x^2-3) = 72 oh nvm 2 was right

Answers:I don't understand the way you've typed the question. The symbol ^ means "exponent", which doesn't make sense here except where you put x^2. A guess: Does number 1 mean :"log of 30 with base 7"? If so, we can write it here as log(7) 30 or log(base 7) 30 or log 30 to base 7. If it's that, calculate it as (log 30)/()log 7) =1.747869697 While using Excel for that calculation just now, I notice they use log(30,7) to mean log of 30 with base 7. Which makes me think that maybe q.1 is log of 7 in base 30, which is the reciprocal of the answer I've found. Q.2 What you have done would give a correct solution to the equation 5^(x^2-3) = 72 I didn't work through to check your figures, but the process is correct and the numbers look reasonable. Is that the equation you had to solve? I notice there's a 71in the equation you typed for us, but I don't think the answer would be very different whether it was 71 or 72. So tell us, what was the equation really?

Question:the problem is: 16^2x-5= 2^7 ^ = the power of. do you know how to solve this?

Answers:16^(2x-5) = 2^7 (2^4)^(2x-5) = 2^7 2^(8x-20) = 2^7 because the bases are the same you can set the powers equal to each other.. 8x-20 = 7 8x = 27 x = 27/8

Question:4e^-2x=17 I don't understand how to solve it since it has e in it. I know how to do the other ones like 7^6x=12. but that doesn't have e. So could you please explain to me how to work this problem? I did try, and I got -1.0219. But the book says its -.723. So I don't know what I did wrong.. thanks! (:

Answers:Divide both sides by 4: 4e^(-2x)/2 = 17/4 e^(-2x) = 17/4 Then, set both sides by ln: ln(e^(-2x)) = ln(17/4) -2x = ln(17/4) Finally, divide both sides by -2: x = ln(17/4)/-2 Therefore, by calculator, you get x -.723. I hope this helps!

Question:Solve the exponential equation algebraically. Round results to 3 decimal places. 40/(3-1e^-0.0001x) = 1000 The answer is: - 1085.189 The question I am asking is how do i work the problem to get the answer -1085.189 Please show me how to get the answer Thanks

Answers:40 / (3 - e^[-0.0001x]) = 1000 3 - e^[-0.0001x] = 40/100 3 - e^[-0.0001x] = 0.04 e^[-0.0001x] = 3 - 0.04 e^[-0.0001x] = 2.96. . . . . . . . .| ln() both sides ln{e^[-0.0001x]} = ln(2.96) [-0.0001x]*ln(e) = ln(2.96) [-0.0001x]*1 = ln(2.96) -0.0001*x = ln(2.96) x = ln(2.96) / (-0.0001) = - ln(2.96) / 0.0001 aprox= - 1.085189268 / 0.0001 x aprox= - 1085.189 There you have it! Best regards!

From Youtube

Solving Exponential Equations :Just a video showing how to solve for ' x ' in some cases, if ' x ' happens to be an exponent. A few different examples. For more free math videos, visit: PatrickJMT.com

Exponents & Logs: Solve Exponential Equations :www.mindbites.com This 76 minute exponents & logarithms lesson focuses on solving equations with variables as exponents, for example 9^(2x+1) = 81^4 times 27^x: This lesson will show you how to solve exponential equations: - with the same bases - with different bases - with two solutions - using a system of linear equations - by factoring Thislesson contains explanations of the concepts and 35 example questions with step by step solutions plus 6 interactive review questions with solutions. Lessons that will help you with the fundamentals of this lesson include: - 115 The 5 Basic Exponent Laws (www.mindbites.com - 165 The Zero Negative & Rational Exponent (www.mindbites.com - 205 Solving Systems of Linear Equations (www.mindbites.com - 230 Solving Quadratic Equations by Factoring (www.mindbites.com