Solution - Factoring binomials using the difference of squares

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  24p2 -  9  = 0 

Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  16p²-9 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check :  16  is the square of  4 
Check : 9 is the square of 3
Check :  p²  is the square of  p¹ 

Factorization is :       (4p + 3)  •  (4p - 3) 

Equation at the end of step  2  :

  (4p + 3) • (4p - 3)  = 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  :    4p+3 = 0 

 Subtract  3  from both sides of the equation : 
                      4p = -3
Divide both sides of the equation by 4:
                     p = -3/4 = -0.750

Solving a Single Variable Equation :

 3.3      Solve  :    4p-3 = 0 

 Add  3  to both sides of the equation : 
                      4p = 3
Divide both sides of the equation by 4:
                     p = 3/4 = 0.750

Two solutions were found :

1.  p = 3/4 = 0.750
   
2.  p = -3/4 = -0.750