Solution - Reducing fractions to their lowest terms

Rearrange:

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

                     x^2-(49/69)=0 

Step by step solution :

Step  1  :

            49
 Simplify   ——
            69

Equation at the end of step  1  :

          49
  (x2) -  ——  = 0 
          69
 Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  69  as the denominator :

           x2     x2 • 69
     x2 =  ——  =  ———————
           1        69   

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now 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 • 69 - (49)     69x2 - 49
 ——————————————  =  —————————
       69              69    

Trying to factor as a Difference of Squares :

 2.3      Factoring:  69x² - 49 

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 :  69  is not a square !!

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

Equation at the end of step  2  :

  69x2 - 49
  —————————  = 0 
     69    

Step  3  :

When a fraction equals zero :

 3.1    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:

  69x2-49
  ——————— • 69 = 0 • 69
    69   

Now, on the left hand side, the  69  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   69x²-49  = 0

Solving a Single Variable Equation :

 3.2      Solve  :    69x²-49 = 0 

 Add  49  to both sides of the equation : 
                      69x² = 49
Divide both sides of the equation by 69:
                     x² = 49/69 = 0.710
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 49/69  

 The equation has two real solutions  
 These solutions are  x = ±√ 0.710 = ± 0.84270  
 

Two solutions were found :