Solution - Simplifying radicals

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (2x2 +  12) +  36  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   2x² + 48  =   2 • (x² + 24) 

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(x) = x² + 24
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  24.

 The factor(s) are:

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

 Let us test ....

Polynomial Roots Calculator found no rational roots

Equation at the end of step  3  :

  2 • (x2 + 24)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    2   =  0

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

Solving a Single Variable Equation :

 4.2      Solve  :    x²+24 = 0 

 Subtract  24  from both sides of the equation : 
                      x² = -24
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ -24  

 In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 

Accordingly,  √ -24  =
                    √ -1• 24   =
                    √ -1 •√  24   =
                    i •  √ 24

Can  √ 24 be simplified ?

Yes!   The prime factorization of  24   is
   2•2•2•3 
To be able to remove something from under the radical, there have to be  2  instances of it (because we are taking a square i.e. second root).

√ 24   =  √ 2•2•2•3   =
                ±  2 • √ 6

The equation has no real solutions. It has 2 imaginary, or complex solutions.

                      x=  0.0000 + 4.8990 i 
                      x=  0.0000 - 4.8990 i 

Two solutions were found :

1.   x=  0.0000 - 4.8990 i 
   
2.   x=  0.0000 + 4.8990 i