SOLUTION: Hello. Can you please help me fully factorize these: 1. 3x^2-3(x+2)^2 2. 12x^2-27(3+x)^2 Thank you so much!

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Hello. Can you please help me fully factorize these: 1. 3x^2-3(x+2)^2 2. 12x^2-27(3+x)^2 Thank you so much!      Log On


   

Question 466681: Hello. Can you please help me fully factorize these:
1. 3x^2-3(x+2)^2
2. 12x^2-27(3+x)^2
Thank you so much!
Found 2 solutions by mananth, Edwin McCravy:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
3x%5E2-3%28x%2B2%29%5E2
3%28x%5E2-%28x%2B2%29%5E2%29
difference of two squares
3%28%28x%2Bx-2%29%28x-x-2%29%29
3%28%282x-2%29%28-2%29%29
%283%282%28x-2%29%28-2%29%29%29
3*2*(-2)(x-2)
-12(x-2)

2. 12x^2-27(3+x)^2
3%284x%5E2-9%283%2Bx%29%5E2%29
3%28%282x%29%5E2-%283%5E2%29%283-x%29%5E2%29%29
3%28%282x%2B9-3x%29%28%282x-9%2B3x%29%29%29
%283%289-x%29%285x-9%29%29

Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!
You're studying factoring the difference of two squares

    A² - B² = (A - B)(A + B)

-------------------------------------

1.    3x² - 3(x+2)²

First take out the common factor 3, using []'s

  3[x² - (x+2)²]   

Now factor the expression in the []'s as the difference 
of two squares:

 3[x - (x+2)][x + (x+2)]

Simplify:

 3[x - x - 2][x + x + 2]

 3[-2][2x+2]

Multiply the 3 by the -2 and also take out common
factor 2

   -6[2(x+1)] 

    -12(x+1)

That's it.

---------------------------------

2.    12x²-27(3+x)²


   First take out the common factor 3, using []'s

  3[4x² - 9(3+x)²]   

Now factor the expression in the []'s as the difference 
of two squares:

 3[2x - 3(3+x)][2x + 3(3+x)]

Simplify:

 3[2x - 9 - 3x][2x + 9 + 3x]

 3[-x - 9][5x + 9]

Take out common factor -1 in [-x - 9]

 3(-1)[x + 9][5x + 9]
 
   -3(x + 9)(5x + 9)

That's it.

Edwin