Solution - Adding, subtracting and finding the least common multiple

Rearrange:

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

                4/x-7+7/x+7-(-56/x^2-49)=0 

Step by step solution :

Step  1  :

            56
 Simplify   ——
            x2

Equation at the end of step  1  :

     4    7         56
  (((—-7)+—)+7)-((0-——)-49)  = 0 
     x    x         x2

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  x²  as the denominator :

          49     49 • x2
    49 =  ——  =  ———————
          1        x2   

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:

 -56 - (49 • x2)     -49x2 - 56
 ———————————————  =  ——————————
       x2                x2    

Equation at the end of step  2  :

     4    7     (-49x2-56)
  (((—-7)+—)+7)-——————————  = 0 
     x    x         x2    

Step  3  :

            7
 Simplify   —
            x

Equation at the end of step  3  :

     4          7           (-49x2 - 56)
  (((— -  7) +  —) +  7) -  ————————————  = 0 
     x          x                x2     

Step  4  :

            4
 Simplify   —
            x

Equation at the end of step  4  :

     4          7           (-49x2 - 56)
  (((— -  7) +  —) +  7) -  ————————————  = 0 
     x          x                x2     

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

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

         7     7 • x
    7 =  —  =  —————
         1       x  

Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions

 4 - (7 • x)     4 - 7x
 ———————————  =  ——————
      x            x   

Equation at the end of step  5  :

    (4 - 7x)    7           (-49x2 - 56)
  ((———————— +  —) +  7) -  ————————————  = 0 
       x        x                x2     

Step  6  :

Adding fractions which have a common denominator :

 6.1       Adding fractions which have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 (4-7x) + 7     11 - 7x
 ——————————  =  ———————
     x             x   

Equation at the end of step  6  :

   (11 - 7x)          (-49x2 - 56)
  (————————— +  7) -  ————————————  = 0 
       x                   x2     

Step  7  :

Rewriting the whole as an Equivalent Fraction :

 7.1   Adding a whole to a fraction

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

         7     7 • x
    7 =  —  =  —————
         1       x  

Adding fractions that have a common denominator :

 7.2       Adding up the two equivalent fractions

 (11-7x) + 7 • x     11
 ———————————————  =  ——
        x            x 

Equation at the end of step  7  :

  11    (-49x2 - 56)
  —— -  ————————————  = 0 
  x          x2     

Step  8  :

Step  9  :

Pulling out like terms :

 9.1     Pull out like factors :

   -49x² - 56  =   -7 • (7x² + 8) 

Polynomial Roots Calculator :

 9.2    Find roots (zeroes) of :       F(x) = 7x² + 8
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  7  and the Trailing Constant is  8.

 The factor(s) are:

of the Leading Coefficient :  1,7
 of the Trailing Constant :  1 ,2 ,4 ,8

 Let us test ....

Polynomial Roots Calculator found no rational roots

Calculating the Least Common Multiple :

 9.3    Find the Least Common Multiple

      The left denominator is :       x 

      The right denominator is :       x² 

      Least Common Multiple:
      x² 

Calculating Multipliers :

 9.4    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 = x

   Right_M = L.C.M / R_Deno = 1

Making Equivalent Fractions :

 9.5      Rewrite the two fractions into equivalent fractions

   L. Mult. • L. Num.      11 • x
   ——————————————————  =   ——————
         L.C.M               x2  

   R. Mult. • R. Num.      -7 • (7x2+8)
   ——————————————————  =   ————————————
         L.C.M                  x2     

Adding fractions that have a common denominator :

 9.6       Adding up the two equivalent fractions

 11 • x - (-7 • (7x2+8))     49x2 + 11x + 56
 ———————————————————————  =  ———————————————
           x2                      x2       

Trying to factor by splitting the middle term

 9.7     Factoring  49x² + 11x + 56 

The first term is,  49x²  its coefficient is  49 .
The middle term is,  +11x  its coefficient is  11 .
The last term, "the constant", is  +56 

Step-1 : Multiply the coefficient of the first term by the constant   49 • 56 = 2744 

Step-2 : Find two factors of  2744  whose sum equals the coefficient of the middle term, which is   11 .

For tidiness, printing of 26 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  9  :

  49x2 + 11x + 56
  ———————————————  = 0 
        x2       

Step  10  :

When a fraction equals zero :

 10.1    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:

  49x2+11x+56
  ——————————— • x2 = 0 • x2
      x2     

Now, on the left hand side, the  x²  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   49x²+11x+56  = 0

Parabola, Finding the Vertex :

 10.2      Find the Vertex of   y = 49x²+11x+56

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, 49 , 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  -0.1122  

 Plugging into the parabola formula  -0.1122  for  x  we can calculate the  y -coordinate : 
  y = 49.0 * -0.11 * -0.11 + 11.0 * -0.11 + 56.0
or   y = 55.383

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 49x²+11x+56
Axis of Symmetry (dashed)  {x}={-0.11} 
Vertex at  {x,y} = {-0.11,55.38} 
Function has no real roots

Solve Quadratic Equation by Completing The Square

 10.3     Solving   49x²+11x+56 = 0 by Completing The Square .

 Divide both sides of the equation by  49  to have 1 as the coefficient of the first term :
   x²+(11/49)x+(8/7) = 0

Subtract  8/7  from both side of the equation :
   x²+(11/49)x = -8/7

Now the clever bit: Take the coefficient of  x , which is  11/49 , divide by two, giving  11/98 , and finally square it giving  121/9604 

Add  121/9604  to both sides of the equation :
  On the right hand side we have :
   -8/7  +  121/9604   The common denominator of the two fractions is  9604   Adding  (-10976/9604)+(121/9604)  gives  -10855/9604 
  So adding to both sides we finally get :
   x²+(11/49)x+(121/9604) = -10855/9604

Adding  121/9604  has completed the left hand side into a perfect square :
   x²+(11/49)x+(121/9604)  =
   (x+(11/98)) • (x+(11/98))  =
  (x+(11/98))²
Things which are equal to the same thing are also equal to one another. Since
   x²+(11/49)x+(121/9604) = -10855/9604 and
   x²+(11/49)x+(121/9604) = (x+(11/98))²
then, according to the law of transitivity,
   (x+(11/98))² = -10855/9604

We'll refer to this Equation as  Eq. #10.3.1  

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

Note that the square root of
   (x+(11/98))²   is
   (x+(11/98))^2/2 =
  (x+(11/98))¹ =
   x+(11/98)

Now, applying the Square Root Principle to  Eq. #10.3.1  we get:
   x+(11/98) = √ -10855/9604

Subtract  11/98  from both sides to obtain:
   x = -11/98 + √ -10855/9604
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 + (11/49)x + (8/7) = 0
   has two solutions:
  x = -11/98 + √ 10855/9604 •  i 
   or
  x = -11/98 - √ 10855/9604 •  i 

Note that  √ 10855/9604 can be written as
  √ 10855  / √ 9604   which is √ 10855  / 98

Solve Quadratic Equation using the Quadratic Formula

 10.4     Solving    49x²+11x+56 = 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   =     49
                      B   =    11
                      C   =   56

Accordingly,  B²  -  4AC   =
                     121 - 10976 =
                     -10855

Applying the quadratic formula :

               -11 ± √ -10855
   x  =    ————————
                        98

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,√ -10855  = 
                    √ 10855 • (-1)  =
                    √ 10855  • √ -1   =
                    ±  √ 10855  • i

  √ 10855   , rounded to 4 decimal digits, is  104.1873
 So now we are looking at:
           x  =  ( -11 ±  104.187 i ) / 98

Two imaginary solutions :

 x =(-11+√-10855)/98=(-11+i√ 10855 )/98= -0.1122+1.0631i
  or: 
 x =(-11-√-10855)/98=(-11-i√ 10855 )/98= -0.1122-1.0631i

Two solutions were found :

1.  x =(-11-√-10855)/98=(-11-i√ 10855 )/98= -0.1122-1.0631i
2.  x =(-11+√-10855)/98=(-11+i√ 10855 )/98= -0.1122+1.0631i