Solution - Equations reducible to quadratic form

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (3x26 • x) -  24  = 0 
 Step  2  :
 Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   3x²⁷ - 24  =   3 • (x²⁷ - 8) 

Trying to factor as a Difference of Cubes:

 3.2      Factoring:  x²⁷ - 8 

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 :  8  is the cube of   2 
Check :  x²⁷ is the cube of   x⁹

Factorization is :
             (x⁹ - 2)  •  (x¹⁸ + 2x⁹ + 4) 

Trying to factor as a Difference of Cubes:

 3.3      Factoring:  x⁹ - 2 

Check :  2  is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes

Polynomial Roots Calculator :

 3.4    Find roots (zeroes) of :       F(x) = x⁹ - 2
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  1  and the Trailing Constant is  -2.

 The factor(s) are:

of the Leading Coefficient :  1
 of the Trailing Constant :  1 ,2

 Let us test ....

Polynomial Roots Calculator found no rational roots

Trying to factor by splitting the middle term

 3.5     Factoring  x¹⁸ + 2x⁹ + 4 

The first term is,  x¹⁸  its coefficient is  1 .
The middle term is,  +2x⁹  its coefficient is  2 .
The last term, "the constant", is  +4 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 4 = 4 

Step-2 : Find two factors of  4  whose sum equals the coefficient of the middle term, which is   2 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  3  :

  3 • (x9 - 2) • (x18 + 2x9 + 4)  = 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.

Equations which are never true :

 4.2      Solve :    3   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

 4.3      Solve  :    x⁹-2 = 0 

 Add  2  to both sides of the equation : 
                      x⁹ = 2
                     x  =  9th root of (2) 

 The equation has one real solution
This solution is  x = 9th root of 2 = 1.0801

Solving a Single Variable Equation :

Equations which are reducible to quadratic :

 4.4     Solve   x¹⁸+2x⁹+4 = 0

This equation is reducible to quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using  w , such that  w = x⁹  transforms the equation into :
 w²+2w+4 = 0

Solving this new equation using the quadratic formula we get two imaginary solutions :
   w = -1.0000 ± 1.7321 i 
Now that we know the value(s) of  w , we can calculate  x  since  x  is the 9 root of   w  

Since we are speaking 9th root, each of the two imaginary solutions of has 9 roots

Tiger finds these roots using de Moivre's Formula

The 9th roots of  -1.000 + 1.732 i   are:

9th roots of  -1.000- 1.732 i  :
 

19 solutions were found :