Solution - Factoring binomials using the difference of squares

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x6"   was replaced by   "x^6". 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (x2) -  22x6  = 0 
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   x² - 4x⁶  =   -x² • (4x⁴ - 1) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  4x⁴ - 1 

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 : 1 is the square of 1
Check :  x⁴  is the square of  x² 

Factorization is :       (2x² + 1)  •  (2x² - 1) 

Polynomial Roots Calculator :

 3.3    Find roots (zeroes) of :       F(x) = 2x² + 1
Polynomial Roots Calculator is a set of methods aimed at finding values of  x  for which   F(x)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  x  which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number  P/Q   then  P  is a factor of the Trailing Constant and  Q  is a factor of the Leading Coefficient

In this case, the Leading Coefficient is  2  and the Trailing Constant is  1.

 The factor(s) are:

of the Leading Coefficient :  1,2
 of the Trailing Constant :  1

 Let us test ....

Polynomial Roots Calculator found no rational roots

Trying to factor as a Difference of Squares :

 3.4      Factoring:  2x² - 1 

Check :  2  is not a square !!

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

Equation at the end of step  3  :

  -x2 • (2x2 + 1) • (2x2 - 1)  = 0 

Step  4  :

Theory - Roots of a product :

 4.1    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 :

 4.2      Solve  :    -x² = 0 

 Multiply both sides of the equation by (-1) :  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 :

 4.3      Solve  :    2x²+1 = 0 

 Subtract  1  from both sides of the equation : 
                      2x² = -1
Divide both sides of the equation by 2:
                     x² = -1/2 = -0.500
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ -1/2  

 In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 

Accordingly,  √ -1/2  =
                    √ -1• 1/2   =
                    √ -1 •√  1/2   =
                    i •  √ 1/2

The equation has no real solutions. It has 2 imaginary, or complex solutions.

                      x=  0.0000 + 0.7071 i 
                      x=  0.0000 - 0.7071 i 

Solving a Single Variable Equation :

 4.4      Solve  :    2x²-1 = 0 

 Add  1  to both sides of the equation : 
                      2x² = 1
Divide both sides of the equation by 2:
                     x² = 1/2 = 0.500
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 1/2  

 The equation has two real solutions  
 These solutions are  x = ±√ 0.500 = ± 0.70711  
 

5 solutions were found :

1.  x = ±√ 0.500 = ± 0.70711
2.   x=  0.0000 - 0.7071 i 
   
3.   x=  0.0000 + 0.7071 i 
   
4.  x = 0