Solution - Quadratic equations

Rearrange:

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

                     56*x-15-(56/x)=0 

Step by step solution :

Step  1  :

            56
 Simplify   ——
            x 

Equation at the end of step  1  :

                56
  (56x - 15) -  ——  = 0 
                x 

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

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

                 56x - 15     (56x - 15) • x
     56x - 15 =  ————————  =  ——————————————
                    1               x       

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:

 (56x-15) • x - (56)     56x2 - 15x - 56
 ———————————————————  =  ———————————————
          x                     x       

Trying to factor by splitting the middle term

 2.3     Factoring  56x² - 15x - 56 

The first term is,  56x²  its coefficient is  56 .
The middle term is,  -15x  its coefficient is  -15 .
The last term, "the constant", is  -56 

Step-1 : Multiply the coefficient of the first term by the constant   56 • -56 = -3136 

Step-2 : Find two factors of  -3136  whose sum equals the coefficient of the middle term, which is   -15 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -64  and  49 
                     56x² - 64x + 49x - 56

Step-4 : Add up the first 2 terms, pulling out like factors :
                    8x • (7x-8)
              Add up the last 2 terms, pulling out common factors :
                    7 • (7x-8)
Step-5 : Add up the four terms of step 4 :
                    (8x+7)  •  (7x-8)
             Which is the desired factorization

Equation at the end of step  2  :

  (7x - 8) • (8x + 7)
  ———————————————————  = 0 
           x         

Step  3  :

When a fraction equals zero :

 3.1    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:

  (7x-8)•(8x+7)
  ————————————— • x = 0 • x
        x      

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

The equation now takes the shape :
   (7x-8)  •  (8x+7)  = 0

Theory - Roots of a product :

 3.2    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation :

 3.3      Solve  :    7x-8 = 0 

 Add  8  to both sides of the equation : 
                      7x = 8
Divide both sides of the equation by 7:
                     x = 8/7 = 1.143

Solving a Single Variable Equation :

 3.4      Solve  :    8x+7 = 0 

 Subtract  7  from both sides of the equation : 
                      8x = -7
Divide both sides of the equation by 8:
                     x = -7/8 = -0.875

Supplement : Solving Quadratic Equation Directly

Solving    56x2-15x-56  = 0   directly 

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex :

 4.1      Find the Vertex of   y = 56x²-15x-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, 56 , 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   0.1339  

 Plugging into the parabola formula   0.1339  for  x  we can calculate the  y -coordinate : 
  y = 56.0 * 0.13 * 0.13 - 15.0 * 0.13 - 56.0
or   y = -57.004

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 56x²-15x-56
Axis of Symmetry (dashed)  {x}={ 0.13} 
Vertex at  {x,y} = { 0.13,-57.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-0.88, 0.00} 
Root 2 at  {x,y} = { 1.14, 0.00} 

Solve Quadratic Equation by Completing The Square

 4.2     Solving   56x²-15x-56 = 0 by Completing The Square .

 Divide both sides of the equation by  56  to have 1 as the coefficient of the first term :
   x²-(15/56)x-1 = 0

Add  1  to both side of the equation :
   x²-(15/56)x = 1

Now the clever bit: Take the coefficient of  x , which is  15/56 , divide by two, giving  15/112 , and finally square it giving  225/12544 

Add  225/12544  to both sides of the equation :
  On the right hand side we have :
   1  +  225/12544    or,  (1/1)+(225/12544) 
  The common denominator of the two fractions is  12544   Adding  (12544/12544)+(225/12544)  gives  12769/12544 
  So adding to both sides we finally get :
   x²-(15/56)x+(225/12544) = 12769/12544

Adding  225/12544  has completed the left hand side into a perfect square :
   x²-(15/56)x+(225/12544)  =
   (x-(15/112)) • (x-(15/112))  =
  (x-(15/112))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(15/56)x+(225/12544) = 12769/12544 and
   x²-(15/56)x+(225/12544) = (x-(15/112))²
then, according to the law of transitivity,
   (x-(15/112))² = 12769/12544

We'll refer to this Equation as  Eq. #4.2.1  

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

Note that the square root of
   (x-(15/112))²   is
   (x-(15/112))^2/2 =
  (x-(15/112))¹ =
   x-(15/112)

Now, applying the Square Root Principle to  Eq. #4.2.1  we get:
   x-(15/112) = √ 12769/12544

Add  15/112  to both sides to obtain:
   x = 15/112 + √ 12769/12544

Since a square root has two values, one positive and the other negative
   x² - (15/56)x - 1 = 0
   has two solutions:
  x = 15/112 + √ 12769/12544
   or
  x = 15/112 - √ 12769/12544

Note that  √ 12769/12544 can be written as
  √ 12769  / √ 12544   which is 113 / 112

Solve Quadratic Equation using the Quadratic Formula

 4.3     Solving    56x²-15x-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   =     56
                      B   =   -15
                      C   =  -56

Accordingly,  B²  -  4AC   =
                     225 - (-12544) =
                     12769

Applying the quadratic formula :

               15 ± √ 12769
   x  =    ———————
                        112

Can  √ 12769 be simplified ?

Yes!   The prime factorization of  12769   is
   113•113 
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).

√ 12769   =  √ 113•113   =
                ±  113 • √ 1   =
                ±  113

So now we are looking at:
           x  =  ( 15 ± 113) / 112

Two real solutions:

x =(15+√12769)/112=(15+113)/112= 1.143

or:

x =(15-√12769)/112=(15-113)/112= -0.875

Two solutions were found :

1.  x = -7/8 = -0.875
   
2.  x = 8/7 = 1.143