Solution - Factoring binomials using the difference of squares

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x4"   was replaced by   "x^4". 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (9 • (x2)) -  (22•3x4)  = 0 
 Step  2  :

Equation at the end of step  2  :

  32x2 -  (22•3x4)  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   9x² - 12x⁴  =   -3x² • (4x² - 3) 

Trying to factor as a Difference of Squares :

 4.2      Factoring:  4x² - 3 

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 :  4  is the square of  2 
Check : 3 is not a square !!

Ruling : Binomial can not be factored as the difference of two perfect squares.

Equation at the end of step  4  :

  -3x2 • (4x2 - 3)  = 0 

Step  5  :

Theory - Roots of a product :

 5.1    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 :

 5.2      Solve  :    -3x² = 0 

 Multiply both sides of the equation by (-1) :  3x² = 0


Divide both sides of the equation by 3:
                     x² = 0
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 0  

 Any root of zero is zero. This equation has one solution which is  x = 0

Solving a Single Variable Equation :

 5.3      Solve  :    4x²-3 = 0 

 Add  3  to both sides of the equation : 
                      4x² = 3
Divide both sides of the equation by 4:
                     x² = 3/4 = 0.750
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 3/4  

 The equation has two real solutions  
 These solutions are  x = ±√ 0.750 = ± 0.86603  
 

Three solutions were found :

1.  x = ±√ 0.750 = ± 0.86603
2.  x = 0