Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "/-4h" was replaced by "/(-4h)".

Rearrange:

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

                     21/(-4*h)-(14/11+h)=0 

Step by step solution :

Step  1  :

            14
 Simplify   ——
            11

Equation at the end of step  1  :

   21     14    
  ——— -  (—— +  h)  = 0 
  -4h     11    

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  11  as the denominator :

         h     h • 11
    h =  —  =  ——————
         1       11  

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:

 14 + h • 11     11h + 14
 ———————————  =  ————————
     11             11   

Equation at the end of step  2  :

   21    (11h + 14)
  ——— -  ——————————  = 0 
  -4h        11    

Step  3  :

             21
 Simplify   ———
            -4h

Equation at the end of step  3  :

   21    (11h + 14)
  ——— -  ——————————  = 0 
  -4h        11    

Step  4  :

Calculating the Least Common Multiple :

 4.1    Find the Least Common Multiple

      The left denominator is :       -4h 

      The right denominator is :       11 

      Least Common Multiple:
      44h 

Calculating Multipliers :

 4.2    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 = -11

   Right_M = L.C.M / R_Deno = 4h

Making Equivalent Fractions :

 4.3      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.      21 • -11
   ——————————————————  =   ————————
         L.C.M               44h   

   R. Mult. • R. Num.      (11h+14) • 4h
   ——————————————————  =   —————————————
         L.C.M                  44h     

Adding fractions that have a common denominator :

 4.4       Adding up the two equivalent fractions

 21 • -11 - ((11h+14) • 4h)     -44h2 - 56h - 231
 ——————————————————————————  =  —————————————————
            44h                        44h       

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   -44h² - 56h - 231  =   -1 • (44h² + 56h + 231) 

Trying to factor by splitting the middle term

 5.2     Factoring  44h² + 56h + 231 

The first term is,  44h²  its coefficient is  44 .
The middle term is,  +56h  its coefficient is  56 .
The last term, "the constant", is  +231 

Step-1 : Multiply the coefficient of the first term by the constant   44 • 231 = 10164 

Step-2 : Find two factors of  10164  whose sum equals the coefficient of the middle term, which is   56 .

For tidiness, printing of 66 lines which failed to find two such factors, was suppressed

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

Equation at the end of step  5  :

  -44h2 - 56h - 231
  —————————————————  = 0 
         44h       

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:

  -44h2-56h-231
  ————————————— • 44h = 0 • 44h
       44h     

Now, on the left hand side, the  44h  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -44h²-56h-231  = 0

Parabola, Finding the Vertex :

 6.2      Find the Vertex of   y = -44h²-56h-231

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens down and accordingly has a highest point (AKA absolute maximum) .    We know this even before plotting  "y"  because the coefficient of the first term, -44 , is negative (smaller 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,Ah²+Bh+C,the  h -coordinate of the vertex is given by  -B/(2A) . In our case the  h  coordinate is  -0.6364  

 Plugging into the parabola formula  -0.6364  for  h  we can calculate the  y -coordinate : 
  y = -44.0 * -0.64 * -0.64 - 56.0 * -0.64 - 231.0
or   y = -213.182

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -44h²-56h-231
Axis of Symmetry (dashed)  {h}={-0.64} 
Vertex at  {h,y} = {-0.64,-213.18} 
Function has no real roots

Solve Quadratic Equation by Completing The Square

 6.3     Solving   -44h²-56h-231 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 44h²+56h+231 = 0  Divide both sides of the equation by  44  to have 1 as the coefficient of the first term :
   h²+(14/11)h+(21/4) = 0

Subtract  21/4  from both side of the equation :
   h²+(14/11)h = -21/4

Now the clever bit: Take the coefficient of  h , which is  14/11 , divide by two, giving  7/11 , and finally square it giving  49/121 

Add  49/121  to both sides of the equation :
  On the right hand side we have :
   -21/4  +  49/121   The common denominator of the two fractions is  484   Adding  (-2541/484)+(196/484)  gives  -2345/484 
  So adding to both sides we finally get :
   h²+(14/11)h+(49/121) = -2345/484

Adding  49/121  has completed the left hand side into a perfect square :
   h²+(14/11)h+(49/121)  =
   (h+(7/11)) • (h+(7/11))  =
  (h+(7/11))²
Things which are equal to the same thing are also equal to one another. Since
   h²+(14/11)h+(49/121) = -2345/484 and
   h²+(14/11)h+(49/121) = (h+(7/11))²
then, according to the law of transitivity,
   (h+(7/11))² = -2345/484

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
   (h+(7/11))²   is
   (h+(7/11))^2/2 =
  (h+(7/11))¹ =
   h+(7/11)

Now, applying the Square Root Principle to  Eq. #6.3.1  we get:
   h+(7/11) = √ -2345/484

Subtract  7/11  from both sides to obtain:
   h = -7/11 + √ -2345/484
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
   h² + (14/11)h + (21/4) = 0
   has two solutions:
  h = -7/11 + √ 2345/484 •  i 
   or
  h = -7/11 - √ 2345/484 •  i 

Note that  √ 2345/484 can be written as
  √ 2345  / √ 484   which is √ 2345  / 22

Solve Quadratic Equation using the Quadratic Formula

 6.4     Solving    -44h²-56h-231 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  h  , the solution for   Ah²+Bh+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  h =   ————————
                      2A

  In our case,  A   =    -44
                      B   =   -56
                      C   =  -231

Accordingly,  B²  -  4AC   =
                     3136 - 40656 =
                     -37520

Applying the quadratic formula :

               56 ± √ -37520
   h  =    ————————
                        -88

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,√ -37520  = 
                    √ 37520 • (-1)  =
                    √ 37520  • √ -1   =
                    ±  √ 37520  • i

Can  √ 37520 be simplified ?

Yes!   The prime factorization of  37520   is
   2•2•2•2•5•7•67 
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).

√ 37520   =  √ 2•2•2•2•5•7•67   =2•2•√ 2345   =
                ±  4 • √ 2345

  √ 2345   , rounded to 4 decimal digits, is  48.4252
 So now we are looking at:
           h  =  ( 56 ± 4 •  48.425 i ) / -88

Two imaginary solutions :

 h =(56+√-37520)/-88=7/-11-i/22√ 2345 = -0.6364+2.2011i
  or: 
 h =(56-√-37520)/-88=7/-11+i/22√ 2345 = -0.6364-2.2011i

Two solutions were found :

1.  h =(56-√-37520)/-88=7/-11+i/22√ 2345 = -0.6364-2.2011i
2.  h =(56+√-37520)/-88=7/-11-i/22√ 2345 = -0.6364+2.2011i