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) -  24x64  = 0 
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   x² - 16x⁶⁴  =   -x² • (16x⁶² - 1) 

Trying to factor as a Difference of Squares :

 3.2      Factoring:  16x⁶² - 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 :  16  is the square of  4 
Check : 1 is the square of 1
Check :  x⁶²  is the square of  x³¹ 

Factorization is :       (4x³¹ + 1)  •  (4x³¹ - 1) 

Equation at the end of step  3  :

  -x2 • (4x31 + 1) • (4x31 - 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  :    4x³¹+1 = 0 

 Subtract  1  from both sides of the equation : 
                      4x³¹ = -1
Divide both sides of the equation by 4:
                     x³¹ = -1/4 = -0.250
                     x  =  31st root of (-1/4) 

 Negative numbers have real 31th roots.
 31st root of (-1/4) = ³¹√ -1• 1/4  = ³¹√ -1 • ³¹√ 1/4  =(-1)•³¹√ 1/4 

The equation has one real solution, a negative number This solution is  x = 31st root of (-0.250) = -0.95627

Solving a Single Variable Equation :

 4.4      Solve  :    4x³¹-1 = 0 

 Add  1  to both sides of the equation : 
                      4x³¹ = 1
Divide both sides of the equation by 4:
                     x³¹ = 1/4 = 0.250
                     x  =  31st root of (1/4) 

 The equation has one real solution
This solution is  x = 31st root of ( 0.250) = 0.95627

Three solutions were found :

1.  x = 31st root of ( 0.250) = 0.95627
2.  x = 31st root of (-0.250) = -0.95627
3.  x = 0