Question

Solve This Equation With "Completeing The Square" Methed and Find Its Roots/Zeros.

Answer 1

(3x2 - x) - 4 = 0

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  3x2-x-4 

The first term is,  3x2  its coefficient is  3 .
The middle term is,  -x  its coefficient is  -1 .
The last term, "the constant", is  -4 

Step-1 : Multiply the coefficient of the first term by the constant  3 • -4 = -12 

Step-2 : Find two factors of  -12 whose sum equals the coefficient of the middle term, which is   -1 .

     -12   +   1   =   -11     -6   +   2   =   -4     -4   +   3   =   -1   That's it


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

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (3x-4)
              Add up the last 2 terms, pulling out common factors :
                     1 • (3x-4)
Step-5 : Add up the four terms of step 4 :
                    (x+1)  •  (3x-4)
             Which is the desired factorization

Equation at the end of step  2 :

(3x - 4) • (x + 1) = 0

Step  3  :

Theory - Roots of a product :

 3.1    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 :

 3.2      Solve  :    3x-4 = 0 

 Add  4  to both sides of the equation : 
                      3x = 4 
Divide both sides of the equation by 3:
                     x = 4/3 = 1.333 

Solving a Single Variable Equation :

 3.3      Solve  :    x+1 = 0 

 Subtract  1  from both sides of the equation : 
                      x = -1 

Supplement : Solving Quadratic Equation Directly

Solving  3x2-x-4  = 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