Solution - Adding, subtracting and finding the least common multiple

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "0.15" was replaced by "(15/100)".

Rearrange:

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

                     x-1500/x-((15/100))=0 

Step by step solution :

Step  1  :

             3
 Simplify   ——
            20

Equation at the end of step  1  :

        1500      3
  (x -  ————) -  ——  = 0 
         x       20

Step  2  :

            1500
 Simplify   ————
             x  

Equation at the end of step  2  :

        1500      3
  (x -  ————) -  ——  = 0 
         x       20

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

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

          x     x • x
     x =  —  =  —————
          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 :

 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:

 x • x - (1500)     x2 - 1500
 ——————————————  =  —————————
       x                x    

Equation at the end of step  3  :

  (x2 - 1500)     3
  ——————————— -  ——  = 0 
       x         20

Step  4  :

Trying to factor as a Difference of Squares :

 4.1      Factoring:  x²-1500 

Theory : A difference of two perfect squares,  A² - B²  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A² - AB + BA - B² =
         A² - AB + AB - B² =
         A² - B²

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 1500 is not a square !!

Ruling : Binomial can not be factored as the difference of two perfect squares.

Calculating the Least Common Multiple :

 4.2    Find the Least Common Multiple

      The left denominator is :       x 

      The right denominator is :       20 

      Least Common Multiple:
      20x 

Calculating Multipliers :

 4.3    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 = 20

   Right_M = L.C.M / R_Deno = x

Making Equivalent Fractions :

 4.4      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.      (x2-1500) • 20
   ——————————————————  =   ——————————————
         L.C.M                  20x      

   R. Mult. • R. Num.      3 • x
   ——————————————————  =   —————
         L.C.M              20x 

Adding fractions that have a common denominator :

 4.5       Adding up the two equivalent fractions

 (x2-1500) • 20 - (3 • x)     20x2 - 3x - 30000
 ————————————————————————  =  —————————————————
           20x                       20x       

Trying to factor by splitting the middle term

 4.6     Factoring  20x² - 3x - 30000 

The first term is,  20x²  its coefficient is  20 .
The middle term is,  -3x  its coefficient is  -3 .
The last term, "the constant", is  -30000 

Step-1 : Multiply the coefficient of the first term by the constant   20 • -30000 = -600000 

Step-2 : Find two factors of  -600000  whose sum equals the coefficient of the middle term, which is   -3 .

For tidiness, printing of 78 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  4  :

  20x2 - 3x - 30000
  —————————————————  = 0 
         20x       

Step  5  :

When a fraction equals zero :

 5.1    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:

  20x2-3x-30000
  ————————————— • 20x = 0 • 20x
       20x     

Now, on the left hand side, the  20x  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   20x²-3x-30000  = 0

Parabola, Finding the Vertex :

 5.2      Find the Vertex of   y = 20x²-3x-30000

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, 20 , 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   0.0750  

 Plugging into the parabola formula   0.0750  for  x  we can calculate the  y -coordinate : 
  y = 20.0 * 0.07 * 0.07 - 3.0 * 0.07 - 30000.0
or   y = -30000.112

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 20x²-3x-30000
Axis of Symmetry (dashed)  {x}={ 0.07} 
Vertex at  {x,y} = { 0.07,-30000.11} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-38.65, 0.00} 
Root 2 at  {x,y} = {38.80, 0.00} 

Solve Quadratic Equation by Completing The Square

 5.3     Solving   20x²-3x-30000 = 0 by Completing The Square .

 Divide both sides of the equation by  20  to have 1 as the coefficient of the first term :
   x²-(3/20)x-1500 = 0

Add  1500  to both side of the equation :
   x²-(3/20)x = 1500

Now the clever bit: Take the coefficient of  x , which is  3/20 , divide by two, giving  3/40 , and finally square it giving  9/1600 

Add  9/1600  to both sides of the equation :
  On the right hand side we have :
   1500  +  9/1600    or,  (1500/1)+(9/1600) 
  The common denominator of the two fractions is  1600   Adding  (2400000/1600)+(9/1600)  gives  2400009/1600 
  So adding to both sides we finally get :
   x²-(3/20)x+(9/1600) = 2400009/1600

Adding  9/1600  has completed the left hand side into a perfect square :
   x²-(3/20)x+(9/1600)  =
   (x-(3/40)) • (x-(3/40))  =
  (x-(3/40))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(3/20)x+(9/1600) = 2400009/1600 and
   x²-(3/20)x+(9/1600) = (x-(3/40))²
then, according to the law of transitivity,
   (x-(3/40))² = 2400009/1600

We'll refer to this Equation as  Eq. #5.3.1  

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

Note that the square root of
   (x-(3/40))²   is
   (x-(3/40))^2/2 =
  (x-(3/40))¹ =
   x-(3/40)

Now, applying the Square Root Principle to  Eq. #5.3.1  we get:
   x-(3/40) = √ 2400009/1600

Add  3/40  to both sides to obtain:
   x = 3/40 + √ 2400009/1600

Since a square root has two values, one positive and the other negative
   x² - (3/20)x - 1500 = 0
   has two solutions:
  x = 3/40 + √ 2400009/1600
   or
  x = 3/40 - √ 2400009/1600

Note that  √ 2400009/1600 can be written as
  √ 2400009  / √ 1600   which is √ 2400009  / 40

Solve Quadratic Equation using the Quadratic Formula

 5.4     Solving    20x²-3x-30000 = 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   =     20
                      B   =    -3
                      C   =  -30000

Accordingly,  B²  -  4AC   =
                     9 - (-2400000) =
                     2400009

Applying the quadratic formula :

               3 ± √ 2400009
   x  =    ————————
                        40

  √ 2400009   , rounded to 4 decimal digits, is  1549.1962
 So now we are looking at:
           x  =  ( 3 ±  1549.196 ) / 40

Two real solutions:

 x =(3+√2400009)/40=38.805

or:

 x =(3-√2400009)/40=-38.655

Two solutions were found :

1.  x =(3-√2400009)/40=-38.655
2.  x =(3+√2400009)/40=38.805