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How do you know when to handle all three coefficients? or only two in factoring?

Sometimes im told to find 2 numbers that add to 6 and multiply to 8 or I need to find the gcf or I need to find number for all the numbers and the varible too? Like how do you know what to do in what problem? Some start with x^2 some start with 8x^2, its confusing.
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2 Answers

I think your question is How to factor The quadratic:
 
   X^ 2 + 7 X + 10  =0 
 
  always    f(x) = ( X +a ) ( X+b ) = X^2 + ( a +b ) X + ab
 
                                                  Roots , the values of X that makes f( x) =0
                                                   
                                                   (  X+a ) = 0   X1 = -a     ( X+ b) = 0   X2 = -b 
 
   X^2 + ( a + b) X + ab =  X^2 + ( X1 + X2 ) X + X1X2
 
       with a =1
 
        always :   X1 + X2 = - b
                        X1 . X 2 = c
 
         X^2 + 7X + 10 =0
                                          - (X1 + X2) =  7     X1 . X2 = 10  ( 2)
 
 
 
                                            2, 5 are the answer for X1 x2
       then :
 
       X^2 + 7X + 10  = ( X +2 ) ( X +5 ) =0   X = -2   X = -5
 
 
             If  System of equation( 2 ) doesn't have integer solution, then we have to factor by completing
 
               the square.
 
           for a ≠ 1
 
            i.e.
                 2 X^2 + 7X +5 = 0        then -( X 1 + X2 )= 7/2     X1 . X2 =  ( 5 /2)
 
                                                          again answer X1 = -1  X2 =- 5/2
             To factor we break  7 into 2 +5
 
              2 X^2 + 2X + 5X + 5=0
 
               Factoring by grouping :
 
                2X ( X + 1 ) + 5 ( X +1 ) = 0
 
                 ( X +1 ) ( 2X + 5) =0
 
                   X = -1   X = -5/2
               
 
              
                             
Life is full of different problems.
 
Based on our past experiences we try to predict what will happen in the future.
 
We sometimes do that with math models.
 
And since they are different problems the models will be different.
 
The study of mathematics is aimed at creating math models for each of the different problems and then solving the models to predict real life.