Solution - Quadratic equations

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "33.33" was replaced by "(3333/100)".

Step by step solution :

Step  1  :

            3333
 Simplify   ————
            100 

Equation at the end of step  1  :

                 3333
  ((x2) -  x) -  ————  = 0 
                 100 

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  100  as the denominator :

               x2 - x     (x2 - x) • 100
     x2 - x =  ——————  =  ——————————————
                 1             100      

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

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   x² - x  =   x • (x - 1) 

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-1) • 100 - (3333)     100x2 - 100x - 3333
 ————————————————————————  =  ———————————————————
           100                        100        

Trying to factor by splitting the middle term

 3.3     Factoring  100x² - 100x - 3333 

The first term is,  100x²  its coefficient is  100 .
The middle term is,  -100x  its coefficient is  -100 .
The last term, "the constant", is  -3333 

Step-1 : Multiply the coefficient of the first term by the constant   100 • -3333 = -333300 

Step-2 : Find two factors of  -333300  whose sum equals the coefficient of the middle term, which is   -100 .

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  3  :

  100x2 - 100x - 3333
  ———————————————————  = 0 
          100        

Step  4  :

When a fraction equals zero :

 4.1    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:

  100x2-100x-3333
  ——————————————— • 100 = 0 • 100
        100      

Now, on the left hand side, the  100  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   100x²-100x-3333  = 0

Parabola, Finding the Vertex :

 4.2      Find the Vertex of   y = 100x²-100x-3333

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, 100 , 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   0.5000  

 Plugging into the parabola formula   0.5000  for  x  we can calculate the  y -coordinate : 
  y = 100.0 * 0.50 * 0.50 - 100.0 * 0.50 - 3333.0
or   y = -3358.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 100x²-100x-3333
Axis of Symmetry (dashed)  {x}={ 0.50} 
Vertex at  {x,y} = { 0.50,-3358.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-5.29, 0.00} 
Root 2 at  {x,y} = { 6.29, 0.00} 

Solve Quadratic Equation by Completing The Square

 4.3     Solving   100x²-100x-3333 = 0 by Completing The Square .

 Divide both sides of the equation by  100  to have 1 as the coefficient of the first term :
   x²-x-(3333/100) = 0

Add  3333/100  to both side of the equation :
   x²-x = 3333/100

Now the clever bit: Take the coefficient of  x , which is  1 , divide by two, giving  1/2 , and finally square it giving  1/4 

Add  1/4  to both sides of the equation :
  On the right hand side we have :
   3333/100  +  1/4   The common denominator of the two fractions is  100   Adding  (3333/100)+(25/100)  gives  3358/100 
  So adding to both sides we finally get :
   x²-x+(1/4) = 1679/50

Adding  1/4  has completed the left hand side into a perfect square :
   x²-x+(1/4)  =
   (x-(1/2)) • (x-(1/2))  =
  (x-(1/2))²
Things which are equal to the same thing are also equal to one another. Since
   x²-x+(1/4) = 1679/50 and
   x²-x+(1/4) = (x-(1/2))²
then, according to the law of transitivity,
   (x-(1/2))² = 1679/50

We'll refer to this Equation as  Eq. #4.3.1  

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

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

Now, applying the Square Root Principle to  Eq. #4.3.1  we get:
   x-(1/2) = √ 1679/50

Add  1/2  to both sides to obtain:
   x = 1/2 + √ 1679/50

Since a square root has two values, one positive and the other negative
   x² - x - (3333/100) = 0
   has two solutions:
  x = 1/2 + √ 1679/50
   or
  x = 1/2 - √ 1679/50

Note that  √ 1679/50 can be written as
  √ 1679  / √ 50 

Solve Quadratic Equation using the Quadratic Formula

 4.4     Solving    100x²-100x-3333 = 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   =    100
                      B   =   -100
                      C   =  -3333

Accordingly,  B²  -  4AC   =
                     10000 - (-1333200) =
                     1343200

Applying the quadratic formula :

               100 ± √ 1343200
   x  =    ————————
                        200

Can  √ 1343200 be simplified ?

Yes!   The prime factorization of  1343200   is
   2•2•2•2•2•5•5•23•73 
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).

√ 1343200   =  √ 2•2•2•2•2•5•5•23•73   =2•2•5•√ 3358   =
                ±  20 • √ 3358

  √ 3358   , rounded to 4 decimal digits, is  57.9483
 So now we are looking at:
           x  =  ( 100 ± 20 •  57.948 ) / 200

Two real solutions:

 x =(100+√1343200)/200=1/2+1/10√ 3358 = 6.295

or:

 x =(100-√1343200)/200=1/2-1/10√ 3358 = -5.295

Two solutions were found :

1.  x =(100-√1343200)/200=1/2-1/10√ 3358 = -5.295
2.  x =(100+√1343200)/200=1/2+1/10√ 3358 = 6.295