Simplify a^2 - 3b^2

Question

Is there a formula for this kind of q?  thank you

Answer 1

 a² - 3b² = ( a - (√3)b)(a + (√3)b)

 

 

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

                                               = X² +aX +bX +ab

                                                =  X^{2 +}(a + b)X +ab   (1)

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

                         = X^{2  +} 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.

          

Answer 2

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