The solution of a simple algebraic problem

 

§01 generation Numerical solution

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   Yes An interesting topic , The difference between skill calculation and numerical calculation is shown .

   Known variables a,b The difference between square and a,b The product of the ,  seek a+b Value . The direct impulse of people who have studied mathematics , It is based on ab=3,  obtain b=3/a. To the first expression   You can list about a The equation of .  Solving this equation .  You can get a The value of square   according to a Square is greater than 0,  You can get a Unique value of ,  The value of abandonment is more complex .  But this is the result a The value of .

  
   Next, according to b=3/a,  You can know b The value of ,  Finally get a+b The value of .  Simplifying this expression is a nightmare .  Here are a,b,a+b Numerical solution of .
  
   This solution process is a standard algebraic solution ,  But the final result of simplification is not very friendly .  Is there any other good method besides direct solution ？

  
   Here is another solution .  For the sake of a+b Value ,  Let's see if we can get (a+b) The square of ？  an a+b The square of , You can see that there are three items ,  In the middle of the ab The product is known ,  but a square +b Square requires .

  
   For the sake of a The square of +b The square of ,  What will be known （a Square minus b square ） Continuous square ,  You can get a The fourth power of , subtract 2 Times a Fang b Fang , add b The fourth power of .  According to known conditions , It's equal to 9 The square of .  Add the equation left and right at the same time 4bz Of a Fang b Fang ,  Left got a Canada b Square term of square .  The right side can be calculated to be equal to 117.

  
   And that gives us a Canada b The value of the square ,  Through both sides at the same time square ,  You can get a Canada b The value of the square .   Will find a Canada b Square into the initial expression ,  You can get a Add b The value of square .  Square off both sides ,  You can get a+b The value of .

  
   Compare the two methods to solve ,  You can see that their results are the same .  The first method , Start simple , The later simplification is complex .  The second method , Need to use math tricks ,  But the result is easier to simplify .  Do you think the internal method is better ？

  
 

※ total    junction ※

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   Ben This paper tests the use of PAD Complete the process of solving algebraic equations .

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