Solution - Quadratic equations

Step by step solution :

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  p²-2000p+350000 

The first term is,  p²  its coefficient is  1 .
The middle term is,  -2000p  its coefficient is  -2000 .
The last term, "the constant", is  +350000 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 350000 = 350000 

Step-2 : Find two factors of  350000  whose sum equals the coefficient of the middle term, which is   -2000 .

For tidiness, printing of 114 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  1  :

  p2 - 2000p + 350000  = 0 

Step  2  :

Parabola, Finding the Vertex :

 2.1      Find the Vertex of   y = p²-2000p+350000

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, 1 , 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,Ap²+Bp+C,the  p -coordinate of the vertex is given by  -B/(2A) . In our case the  p  coordinate is  1000.0000  

 Plugging into the parabola formula  1000.0000  for  p  we can calculate the  y -coordinate : 
  y = 1.0 * 1000.00 * 1000.00 - 2000.0 * 1000.00 + 350000.0
or   y = -650000.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = p²-2000p+350000
Axis of Symmetry (dashed)  {p}={1000.00} 
Vertex at  {p,y} = {1000.00,-650000.00} 
 p -Intercepts (Roots) :
Root 1 at  {p,y} = {193.77, 0.00} 
Root 2 at  {p,y} = {1806.23, 0.00} 

Solve Quadratic Equation by Completing The Square

 2.2     Solving   p²-2000p+350000 = 0 by Completing The Square .

 Subtract  350000  from both side of the equation :
   p²-2000p = -350000

Now the clever bit: Take the coefficient of  p , which is  2000 , divide by two, giving  1000 , and finally square it giving  1000 

Add  1000  to both sides of the equation :
  On the right hand side we have :
   -350000  +  1000    or,  (-350000/1)+(1000/1) 
  The common denominator of the two fractions is  1   Adding  (-350000/1)+(1000/1)  gives  -349000/1 
  So adding to both sides we finally get :
   p²-2000p+1000 = -349000

Adding  1000  has completed the left hand side into a perfect square :
   p²-2000p+1000  =
   (p-1000) • (p-1000)  =
  (p-1000)²
Things which are equal to the same thing are also equal to one another. Since
   p²-2000p+1000 = -349000 and
   p²-2000p+1000 = (p-1000)²
then, according to the law of transitivity,
   (p-1000)² = -349000

We'll refer to this Equation as  Eq. #2.2.1  

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

Note that the square root of
   (p-1000)²   is
   (p-1000)^2/2 =
  (p-1000)¹ =
   p-1000

Now, applying the Square Root Principle to  Eq. #2.2.1  we get:
   p-1000 = √ -349000

Add  1000  to both sides to obtain:
   p = 1000 + √ -349000
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
   p² - 2000p + 350000 = 0
   has two solutions:
  p = 1000 + √ 349000 •  i 
   or
  p = 1000 - √ 349000 •  i 

Solve Quadratic Equation using the Quadratic Formula

 2.3     Solving    p²-2000p+350000 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  p  , the solution for   Ap²+Bp+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  p =   ————————
                      2A

  In our case,  A   =     1
                      B   =   -2000
                      C   =  350000

Accordingly,  B²  -  4AC   =
                     4000000 - 1400000 =
                     2600000

Applying the quadratic formula :

               2000 ± √ 2600000
   p  =    ——————————
                           2

Can  √ 2600000 be simplified ?

Yes!   The prime factorization of  2600000   is
   2•2•2•2•2•2•5•5•5•5•5•13 
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).

√ 2600000   =  √ 2•2•2•2•2•2•5•5•5•5•5•13   =2•2•2•5•5•√ 65   =
                ±  200 • √ 65

  √ 65   , rounded to 4 decimal digits, is   8.0623
 So now we are looking at:
           p  =  ( 2000 ± 200 •  8.062 ) / 2

Two real solutions:

 p =(2000+√2600000)/2=1000+100√ 65 = 1806.226

or:

 p =(2000-√2600000)/2=1000-100√ 65 = 193.774

Two solutions were found :

1.  p =(2000-√2600000)/2=1000-100√ 65 = 193.774
2.  p =(2000+√2600000)/2=1000+100√ 65 = 1806.226