Solution - Simplifying radicals

Rearrange:

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

                     x^2+421/4-(13*x)=0 

Step by step solution :

Step  1  :

            421
 Simplify   ———
             4 

Equation at the end of step  1  :

           421     
  ((x2) +  ———) -  13x  = 0 
            4      
 Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a fraction to a whole

Rewrite the whole as a fraction using  4  as the denominator :

           x2     x2 • 4
     x2 =  ——  =  ——————
           1        4   

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 • 4 + 421     4x2 + 421
 ————————————  =  —————————
      4               4    

Equation at the end of step  2  :

  (4x2 + 421)    
  ——————————— -  13x  = 0 
       4         

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  4  as the denominator :

           13x     13x • 4
    13x =  ———  =  ———————
            1         4   

Polynomial Roots Calculator :

 3.2    Find roots (zeroes) of :       F(x) = 4x² + 421
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  4  and the Trailing Constant is  421.

 The factor(s) are:

of the Leading Coefficient :  1,2 ,4
 of the Trailing Constant :  1 ,421

 Let us test ....

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 3.3       Adding up the two equivalent fractions

 (4x2+421) - (13x • 4)     4x2 - 52x + 421
 —————————————————————  =  ———————————————
           4                      4       

Trying to factor by splitting the middle term

 3.4     Factoring  4x² - 52x + 421 

The first term is,  4x²  its coefficient is  4 .
The middle term is,  -52x  its coefficient is  -52 .
The last term, "the constant", is  +421 

Step-1 : Multiply the coefficient of the first term by the constant   4 • 421 = 1684 

Step-2 : Find two factors of  1684  whose sum equals the coefficient of the middle term, which is   -52 .

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

Equation at the end of step  3  :

  4x2 - 52x + 421
  ———————————————  = 0 
         4       

Step  4  :

When a fraction equals zero :

 4.1    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:

  4x2-52x+421
  ——————————— • 4 = 0 • 4
       4     

Now, on the left hand side, the  4  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   4x²-52x+421  = 0

Parabola, Finding the Vertex :

 4.2      Find the Vertex of   y = 4x²-52x+421

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 4 , is positive (greater than zero). 

 Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. 

 Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. 

 For any parabola,Ax²+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is   6.5000  

 Plugging into the parabola formula   6.5000  for  x  we can calculate the  y -coordinate : 
  y = 4.0 * 6.50 * 6.50 - 52.0 * 6.50 + 421.0
or   y = 252.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 4x²-52x+421
Axis of Symmetry (dashed)  {x}={ 6.50} 
Vertex at  {x,y} = { 6.50,252.00} 
Function has no real roots

Solve Quadratic Equation by Completing The Square

 4.3     Solving   4x²-52x+421 = 0 by Completing The Square .

 Divide both sides of the equation by  4  to have 1 as the coefficient of the first term :
   x²-13x+(421/4) = 0

Subtract  421/4  from both side of the equation :
   x²-13x = -421/4

Now the clever bit: Take the coefficient of  x , which is  13 , divide by two, giving  13/2 , and finally square it giving  169/4 

Add  169/4  to both sides of the equation :
  On the right hand side we have :
   -421/4  +  169/4   The common denominator of the two fractions is  4   Adding  (-421/4)+(169/4)  gives  -252/4 
  So adding to both sides we finally get :
   x²-13x+(169/4) = -63

Adding  169/4  has completed the left hand side into a perfect square :
   x²-13x+(169/4)  =
   (x-(13/2)) • (x-(13/2))  =
  (x-(13/2))²
Things which are equal to the same thing are also equal to one another. Since
   x²-13x+(169/4) = -63 and
   x²-13x+(169/4) = (x-(13/2))²
then, according to the law of transitivity,
   (x-(13/2))² = -63

We'll refer to this Equation as  Eq. #4.3.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x-(13/2))²   is
   (x-(13/2))^2/2 =
  (x-(13/2))¹ =
   x-(13/2)

Now, applying the Square Root Principle to  Eq. #4.3.1  we get:
   x-(13/2) = √ -63

Add  13/2  to both sides to obtain:
   x = 13/2 + √ -63
In Math,  i  is called the imaginary unit. It satisfies   i²  =-1. Both   i   and   -i   are the square roots of   -1 


Since a square root has two values, one positive and the other negative
   x² - 13x + (421/4) = 0
   has two solutions:
  x = 13/2 + √ 63 •  i 
   or
  x = 13/2 - √ 63 •  i 

Solve Quadratic Equation using the Quadratic Formula

 4.4     Solving    4x²-52x+421 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  x  , the solution for   Ax²+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  x =   ————————
                      2A

  In our case,  A   =     4
                      B   =   -52
                      C   =  421

Accordingly,  B²  -  4AC   =
                     2704 - 6736 =
                     -4032

Applying the quadratic formula :

               52 ± √ -4032
   x  =    ———————
                        8

In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written  (a+b*i) 

Both   i   and   -i   are the square roots of minus 1

Accordingly,√ -4032  = 
                    √ 4032 • (-1)  =
                    √ 4032  • √ -1   =
                    ±  √ 4032  • i

Can  √ 4032 be simplified ?

Yes!   The prime factorization of  4032   is
   2•2•2•2•2•2•3•3•7 
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).

√ 4032   =  √ 2•2•2•2•2•2•3•3•7   =2•2•2•3•√ 7   =
                ±  24 • √ 7

  √ 7   , rounded to 4 decimal digits, is   2.6458
 So now we are looking at:
           x  =  ( 52 ± 24 •  2.646 i ) / 8

Two imaginary solutions :

 x =(52+√-4032)/8=13/2+3i√ 7 = 6.5000+7.9373i
  or: 
 x =(52-√-4032)/8=13/2-3i√ 7 = 6.5000-7.9373i

Two solutions were found :

1.  x =(52-√-4032)/8=13/2-3i√ 7 = 6.5000-7.9373i
2.  x =(52+√-4032)/8=13/2+3i√ 7 = 6.5000+7.9373i