Solution - Factoring binomials using the difference of squares

Reformatting the input :

Changes made to your input should not affect the solution:

 (1): "x2"   was replaced by   "x^2". 

(2): "3.4" was replaced by "(34/10)". 3 more similar replacement(s)

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                (2/10)*x^2-(12/10)-((34/10))=0 

Step by step solution :

Step  1  :

            17
 Simplify   ——
            5 

Equation at the end of step  1  :

     2       12  17
  ((——•(x2))-——)-——  = 0 
    10       10  5 

Step  2  :

            6
 Simplify   —
            5

Equation at the end of step  2  :

     2            6     17
  ((—— • (x2)) -  —) -  ——  = 0 
    10            5     5 
 Step  3  :
            1
 Simplify   —
            5

Equation at the end of step  3  :

    1          6     17
  ((— • x2) -  —) -  ——  = 0 
    5          5     5 

Step  4  :

Equation at the end of step  4  :

   x2    6     17
  (—— -  —) -  ——  = 0 
   5     5     5 

Step  5  :

Adding fractions which have a common denominator :

 5.1       Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 x2 - (6)     x2 - 6
 ————————  =  ——————
    5           5   

Equation at the end of step  5  :

  (x2 - 6)    17
  ———————— -  ——  = 0 
     5        5 

Step  6  :

Trying to factor as a Difference of Squares :

 6.1      Factoring:  x²-6 

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 : 6 is not a square !!

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

Adding fractions which have a common denominator :

 6.2       Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 (x2-6) - (17)     x2 - 23
 —————————————  =  ———————
       5              5   

Trying to factor as a Difference of Squares :

 6.3      Factoring:  x² - 23 

Check : 23 is not a square !!

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

Equation at the end of step  6  :

  x2 - 23
  ———————  = 0 
     5   

Step  7  :

When a fraction equals zero :

 7.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  x2-23
  ————— • 5 = 0 • 5
    5  

Now, on the left hand side, the  5  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   x²-23  = 0

Solving a Single Variable Equation :

 7.2      Solve  :    x²-23 = 0 

 Add  23  to both sides of the equation : 
                      x² = 23
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 23  

 The equation has two real solutions  
 These solutions are  x = ± √23 = ± 4.7958  
 

Two solutions were found :