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Adding, subtracting and finding the least common multiple

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3n+2/n+4=8/3

This solution deals with adding, subtracting and finding the least common multiple.

Solution found

n=(-4-sqrt(-200))/18=(-2-5isqrt(2))/9=-0.2222-0.7857i
n=(-4-sqrt(-200))/18=(-2-5isqrt(2))/9=-0.2222-0.7857i
n=(-4+sqrt(-200))/18=(-2+5isqrt(2))/9=-0.2222+0.7857i
n=(-4+sqrt(-200))/18=(-2+5isqrt(2))/9=-0.2222+0.7857i

Step by Step Solution

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Rearrange:

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

                     3*n+2/n+4-(8/3)=0 

Step by step solution :

Step  1  :

            8
 Simplify   —
            3

Equation at the end of step  1  :

          2           8
  ((3n +  —) +  4) -  —  = 0 
          n           3

Step  2  :

            2
 Simplify   —
            n

Equation at the end of step  2  :

          2           8
  ((3n +  —) +  4) -  —  = 0 
          n           3

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a fraction to a whole

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

           3n     3n • n
     3n =  ——  =  ——————
           1        n   

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 :

 3.2       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:

 3n • n + 2     3n2 + 2
 ——————————  =  ———————
     n             n   

Equation at the end of step  3  :

   (3n2 + 2)          8
  (————————— +  4) -  —  = 0 
       n              3

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Adding a whole to a fraction

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

         4     4 • n
    4 =  —  =  —————
         1       n  

Polynomial Roots Calculator :

 4.2    Find roots (zeroes) of :       F(n) = 3n2 + 2
Polynomial Roots Calculator is a set of methods aimed at finding values of  n  for which   F(n)=0  

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers  n  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  3  and the Trailing Constant is  2.

 
The factor(s) are:

of the Leading Coefficient :  1,3
 
of the Trailing Constant :  1 ,2

 
Let us test ....

  P  Q  P/Q  F(P/Q)   Divisor
     -1     1      -1.00      5.00   
     -1     3      -0.33      2.33   
     -2     1      -2.00      14.00   
     -2     3      -0.67      3.33   
     1     1      1.00      5.00   
     1     3      0.33      2.33   
     2     1      2.00      14.00   
     2     3      0.67      3.33   


Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

 4.3       Adding up the two equivalent fractions

 (3n2+2) + 4 • n     3n2 + 4n + 2
 ———————————————  =  ————————————
        n                 n      

Equation at the end of step  4  :

  (3n2 + 4n + 2)    8
  —————————————— -  —  = 0 
        n           3

Step  5  :

Trying to factor by splitting the middle term

 5.1     Factoring  3n2+4n+2 

The first term is,  3n2  its coefficient is  3 .
The middle term is,  +4n  its coefficient is  4 .
The last term, "the constant", is  +2 

Step-1 : Multiply the coefficient of the first term by the constant   3 • 2 = 6 

Step-2 : Find two factors of  6  whose sum equals the coefficient of the middle term, which is   4 .

     -6   +   -1   =   -7
     -3   +   -2   =   -5
     -2   +   -3   =   -5
     -1   +   -6   =   -7
     1   +   6   =   7
     2   +   3   =   5
     3   +   2   =   5
     6   +   1   =   7


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

Calculating the Least Common Multiple :

 5.2    Find the Least Common Multiple

      The left denominator is :       n 

      The right denominator is :       3 

        Number of times each prime factor
        appears in the factorization of:
 Prime 
 Factor 
 Left 
 Denominator 
 Right 
 Denominator 
 L.C.M = Max 
 {Left,Right} 
3011
 Product of all 
 Prime Factors 
133

                  Number of times each Algebraic Factor
            appears in the factorization of:
    Algebraic    
    Factor    
 Left 
 Denominator 
 Right 
 Denominator 
 L.C.M = Max 
 {Left,Right} 
 n 101


      Least Common Multiple:
      3n 

Calculating Multipliers :

 5.3    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 = 3

   Right_M = L.C.M / R_Deno = n

Making Equivalent Fractions :

 5.4      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.      (3n2+4n+2) • 3
   ——————————————————  =   ——————————————
         L.C.M                   3n      

   R. Mult. • R. Num.      8 • n
   ——————————————————  =   —————
         L.C.M              3n  

Adding fractions that have a common denominator :

 5.5       Adding up the two equivalent fractions

 (3n2+4n+2) • 3 - (8 • n)     9n2 + 4n + 6
 ————————————————————————  =  ————————————
            3n                     3n     

Trying to factor by splitting the middle term

 5.6     Factoring  9n2 + 4n + 6 

The first term is,  9n2  its coefficient is  9 .
The middle term is,  +4n  its coefficient is  4 .
The last term, "the constant", is  +6 

Step-1 : Multiply the coefficient of the first term by the constant   9 • 6 = 54 

Step-2 : Find two factors of  54  whose sum equals the coefficient of the middle term, which is   4 .

     -54   +   -1   =   -55
     -27   +   -2   =   -29
     -18   +   -3   =   -21
     -9   +   -6   =   -15
     -6   +   -9   =   -15
     -3   +   -18   =   -21
     -2   +   -27   =   -29
     -1   +   -54   =   -55
     1   +   54   =   55
     2   +   27   =   29
     3   +   18   =   21
     6   +   9   =   15
     9   +   6   =   15
     18   +   3   =   21
     27   +   2   =   29
     54   +   1   =   55


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

Equation at the end of step  5  :

  9n2 + 4n + 6
  ————————————  = 0 
       3n     

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:

  9n2+4n+6
  ———————— • 3n = 0 • 3n
     3n   

Now, on the left hand side, the  3n  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   9n2+4n+6  = 0

Parabola, Finding the Vertex :

 6.2      Find the Vertex of   y = 9n2+4n+6

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, 9 , 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,An2+Bn+C,the  n -coordinate of the vertex is given by  -B/(2A) . In our case the  n  coordinate is  -0.2222  

 
Plugging into the parabola formula  -0.2222  for  n  we can calculate the  y -coordinate : 
 
 y = 9.0 * -0.22 * -0.22 + 4.0 * -0.22 + 6.0
or   y = 5.556

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 9n2+4n+6
Axis of Symmetry (dashed)  {n}={-0.22} 
Vertex at  {n,y} = {-0.22, 5.56} 
Function has no real roots

Solve Quadratic Equation by Completing The Square

 6.3     Solving   9n2+4n+6 = 0 by Completing The Square .

 
Divide both sides of the equation by  9  to have 1 as the coefficient of the first term :
   n2+(4/9)n+(2/3) = 0

Subtract  2/3  from both side of the equation :
   n2+(4/9)n = -2/3

Now the clever bit: Take the coefficient of  n , which is  4/9 , divide by two, giving  2/9 , and finally square it giving  4/81 

Add  4/81  to both sides of the equation :
  On the right hand side we have :
   -2/3  +  4/81   The common denominator of the two fractions is  81   Adding  (-54/81)+(4/81)  gives  -50/81 
  So adding to both sides we finally get :
   n2+(4/9)n+(4/81) = -50/81

Adding  4/81  has completed the left hand side into a perfect square :
   n2+(4/9)n+(4/81)  =
   (n+(2/9)) • (n+(2/9))  =
  (n+(2/9))2
Things which are equal to the same thing are also equal to one another. Since
   n2+(4/9)n+(4/81) = -50/81 and
   n2+(4/9)n+(4/81) = (n+(2/9))2
then, according to the law of transitivity,
   (n+(2/9))2 = -50/81

We'll refer to this Equation as  Eq. #6.3.1  

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

Note that the square root of
   (n+(2/9))2   is
   (n+(2/9))2/2 =
  (n+(2/9))1 =
   n+(2/9)


Now, applying the Square Root Principle to  Eq. #6.3.1  we get:
   n+(2/9) = -50/81

Subtract  2/9  from both sides to obtain:
   n = -2/9 + √ -50/81
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
   n2 + (4/9)n + (2/3) = 0
   has two solutions:
  n = -2/9 + √ 50/81  i 
   or
  n = -2/9 - √ 50/81  i 

Note that  √ 50/81 can be written as
   50  / √ 81   which is  50  / 9

Solve Quadratic Equation using the Quadratic Formula

 6.4     Solving    9n2+4n+6 = 0 by the Quadratic Formula .

 
According to the Quadratic Formula,  n  , the solution for   An2+Bn+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B2-4AC
  n =   ————————
                      2A

  In our case,  A   =     9
                      B   =    4
                      C   =   6

Accordingly,  B2  -  4AC   =
                     16 - 216 =
                     -200

Applying the quadratic formula :

               -4 ± √ -200
   n  =    ——————
                      18

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, -200  = 
                    √ 200 • (-1)  =
                    √ 200  • √ -1   =
                    ±  √ 200  • i


Can  √ 200 be simplified ?

Yes!   The prime factorization of  200   is
   2•2•2•5•5 
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).

200   =  √ 2•2•2•5•5   =2•5•√ 2   =
                ±  10 • √ 2


  √ 2   , rounded to 4 decimal digits, is   1.4142
 So now we are looking at:
           n  =  ( -4 ± 10 •  1.414 i ) / 18

Two imaginary solutions :

 n =(-4+√-200)/18=(-2+5i 2 )/9= -0.2222+0.7857i
  or: 
 n =(-4-√-200)/18=(-2-5i 2 )/9= -0.2222-0.7857i

Two solutions were found :

  1.  n =(-4-√-200)/18=(-2-5i 2 )/9= -0.2222-0.7857i
  2.  n =(-4+√-200)/18=(-2+5i 2 )/9= -0.2222+0.7857i

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