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Please use factoring to solve for x in the following question:

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2 Answers

 
12 X2 - 72X -108 = 0
 
first factor 12:
 
12( X2 - 6X + 9)  =0
 
    Now to factor quadratic,  think of 2 numbers whose sum is 6 and their product 9. Have to do this in your head or scratch paper, or .... the numbers are 3, 3 , 3*3 = 9 , 3+3 =6, then the quadratic factors to:
 
    12 ( X -3 ) 2  = 0     x=3 , is repeated roots of the quadratic
 
     63 X2 - 112 =0
   
       7 ( 9X- 16 ) =0
 
          9 X2 - 16 =0
 
           X2 = 16/9          X =±4/3
 
 ( X + a) ( X + b) = X + ( a +b) X + ab (1)   ( X + a) = X2 + 2aX +a2 (2)   ( X +a ) ( X -a )= X2 -a2 (3) 
 
 The 3 above identities are key to factoring quadratic, and finding the roots
 
    given a quadratic  like :  X2  + 7X + 10
 
 here we see that ( a + b) = 7   ab =10
   a = 2 b =5 is the answer, therefore  X2   + 7X + 10 = ( X +2) ( X + 5 )
   equation (2) is a special case of (1) where a=b , a+b = 2a , ab=a2
   equation (3) is a special case of ( 1) where b = -a, a + ( -a) = 0 , ab = a2
 
  These 3 identities are used in factoring a quadratic.
   
    Equation (1) is factorable if there exists 2 whole number whose sum is ( a+ b), and product =ab.
 
     If the answer of the system of equation is not a whole number, then have to do factoring by competing the square, and come up with a factors of irrational and complex numbers, yielding to Irrational and complex roots . 
  
 
The second one is easy to factor.
 
63x2-112=0 can be factored using property a2-b2=(a-b)*(a+b)
 
63x2-112=7*(9x2-16)=7*(3x-4)*(3x+4). I took factor 7 out to make it look not ugly.
 
So, the equation becomes 7*(3x-4)*(3x+4)=0, from which it follows that x=4/3 or x=-4/3.
 
In the first case,
factor out 12 first.
 
12x2-72x=108 ⇔ 12*(x2-6x)=108 ⇔ x2-6x=9
 
Now move 9 to the left to get:
 
x2-6x-9=0; Now this can be factored as (x-...)*(x-...), the factors in both parentheses having to multiply to -9 and adding to 6. It is not quite obvious how to choose those factors. This is why I despise the factoring as the main way of solving quadratic equations!
 
Here I suggest to add and subtract 9 to the left part. You will get:
 
x2-6x+9-18=0 or x2-6x+9=18
 
Now the expression on the left side can be easily factored, since two factors now have to sum to 6 and multiply to 9. Those are 3 and 3. Thus we get:
 
(x-3)2=18
From here you get:
x-3=±3√2
So,
x1=3+3√2
x2=3-3√2