Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "0.1" was replaced by "(1/10)". 2 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 :

           ((3*x^3)*(x))/(625/100)-((1/10))=0 

Step by step solution :

Step  1  :

             1
 Simplify   ——
            10

Equation at the end of step  1  :

  625  1
  ———-——  = 0 
  100 10

Step  2  :

            25
 Simplify   ——
            4 

Equation at the end of step  2  :

  25     1
  —— -  ——  = 0 
  4     10
 Step  3  :

Equation at the end of step  3  :

  25     1
  —— -  ——  = 0 
  4     10
 Step  4  :
                  25
 Divide  3x4  by  ——
                  4 

Equation at the end of step  4  :

  12x4     1
  ———— -  ——  = 0 
   25     10

Step  5  :

Calculating the Least Common Multiple :

 5.1    Find the Least Common Multiple

      The left denominator is :       25 

      The right denominator is :       10 

      Least Common Multiple:
      50 

Calculating Multipliers :

 5.2    Calculate multipliers for the two fractions


    Denote the Least Common Multiple by  L.C.M 
    Denote the Left Multiplier by  Left_M 
    Denote the Right Multiplier by  Right_M 
    Denote the Left Deniminator by  L_Deno 
    Denote the Right Multiplier by  R_Deno 

   Left_M = L.C.M / L_Deno = 2

   Right_M = L.C.M / R_Deno = 5

Making Equivalent Fractions :

 5.3      Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example :  1/2   and  2/4  are equivalent,  y/(y+1)²   and  (y²+y)/(y+1)³  are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

   L. Mult. • L. Num.      12x4 • 2
   ——————————————————  =   ————————
         L.C.M                50   

   R. Mult. • R. Num.       5
   ——————————————————  =   ——
         L.C.M             50

Adding fractions that have a common denominator :

 5.4       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:

 12x4 • 2 - (5)     24x4 - 5
 ——————————————  =  ————————
       50              50   

Trying to factor as a Difference of Squares :

 5.5      Factoring:  24x⁴ - 5 

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 :  24  is not a square !!

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

Polynomial Roots Calculator :

 5.6    Find roots (zeroes) of :       F(x) = 24x⁴ - 5
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  24  and the Trailing Constant is  -5.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,3 ,4 ,6 ,8 ,12 ,24
 of the Trailing Constant :  1 ,5

 Let us test ....

Note - For tidiness, printing of 27 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

Equation at the end of step  5  :

  24x4 - 5
  ————————  = 0 
     50   

Step  6  :

When a fraction equals zero :

 6.1    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:

  24x4-5
  —————— • 50 = 0 • 50
    50  

Now, on the left hand side, the  50  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   24x⁴-5  = 0

Solving a Single Variable Equation :

 6.2      Solve  :    24x⁴-5 = 0 

 Add  5  to both sides of the equation : 
                      24x⁴ = 5
Divide both sides of the equation by 24:
                     x⁴ = 5/24 = 0.208
                     x  =  ∜ 5/24  

 The equation has two real solutions  
 These solutions are  x = ∜ 0.208 = ± 0.67560  
 

Two solutions were found :