Solution - Linear equations with one unknown

Rearrange:

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

                     r*r^2+48-(97)=0 

Step by step solution :

Step  1  :

Trying to factor as a Difference of Cubes:

 1.1      Factoring:  r³-49 

Theory : A difference of two perfect cubes,  a³ - b³ can be factored into
              (a-b) • (a² +ab +b²)

Proof :  (a-b)•(a²+ab+b²) =
            a³+a²b+ab²-ba²-b²a-b³ =
            a³+(a²b-ba²)+(ab²-b²a)-b³ =
            a³+0+0-b³ =
            a³-b³

Check :  49  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Polynomial Roots Calculator :

 1.2    Find roots (zeroes) of :       F(r) = r³-49
Polynomial Roots Calculator is a set of methods aimed at finding values of  r  for which   F(r)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  r  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  1  and the Trailing Constant is  -49.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,7 ,49

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  1  :

  r3 - 49  = 0 

Step  2  :

Solving a Single Variable Equation :

 2.1      Solve  :    r³-49 = 0 

 Add  49  to both sides of the equation : 
                      r³ = 49
When two things are equal, their cube roots are equal. Taking the cube root of the two sides of the equation we get:  
                      r  =  ∛ 49  

 The equation has one real solution
This solution is  r = ∛49 = 3.6593

One solution was found :