With one method of a procedure called acceptance sampling, a sample of items is randomly selected without replacement and the entire batch is rejected if there is at least one defective unit. The ABCD Electronics Company has just manufactured 6500 write-rewrite CDs, and 110 are defective. If 6 of these 6500 CDs are randomly selected and tested, what is the probability that the entire batch will be rejected

 x2-16x+48=0 Two solutions were found : x = 12 x = 4Step by step solution :Step  1  :Trying to factor by splitting the middle term

 1.1     Factoring  x2-16x+48 

The first term is,  x2  its coefficient is  1 .
The middle term is,  -16x  its coefficient is  -16 .
The last term, "the constant", is  +48 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 48 = 48 

Step-2 : Find two factors of  48  whose sum equals the coefficient of the middle term, which is   -16 .

     -48   +   -1   =   -49     -24   +   -2   =   -26     -16   +   -3   =   -19     -12   +   -4   =   -16   That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -12  and  -4 
                     x2 - 12x - 4x - 48

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-12)
              Add up the last 2 terms, pulling out common factors :
                    4 • (x-12)
Step-5 : Add up the four terms of step 4 :
                    (x-4)  •  (x-12)
             Which is the desired factorization

Equation at the end of step  1  : (x - 4) • (x - 12) = 0 Step  2  :Theory - Roots of a product :

 2.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

 2.2      Solve  :    x-4 = 0 

 Add  4  to both sides of the equation : 
                      x = 4 

Solving a Single Variable Equation :

 2.3      Solve  :    x-12 = 0 

 Add  12  to both sides of the equation : 
                      x = 12 

Supplement : Solving Quadratic Equation DirectlySolving  x2-16x+48  = 0 directly

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = x2-16x+48