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Simplify a^2 - 3b^2

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2 Answers

 a2 - 3b2 = ( a - (√3)b)(a + (√3)b)

 
 

  Note : ( X +a)(X+b) = ( X + a) X + b( X +a) =

                                               = X+aX +bX +ab

                                                =  X2 +(a + b)X +ab   (1)

          (X +a)= ( X +a )(X +a), where a, b = a , a +b = 2a

                         = X2  + 2 aX + a2  (2)

            ( X + a ) ( X -a ) =, where b = -a  , then a - a =0

                                         = X2 - a2    (3)

 

             These 3 identity should be memorized, and should have enough practice on

               them. It comes into great use.

          

This has the format of X^2 + B*Y^2 which can be factored into two terms:
 
(X - SQRT(B)*Y)*(X + SQRT(B)*Y)
 
Thus, this can be rewritten as (a + SQRT(3)*b)*(a - SQRT(3)*b)
 
As a check, multiply this out:
 
a + SQRT(3)*b
a - SQRT(3)*b
-------------------
- a*SQRT(3)*b - 3*b^2
+a*SQRT(3)*b + a^2
 
= a^2 - 3b^2