Solution - Quadratic equations

Step by step solution :

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  x²+20x-15000 

The first term is,  x²  its coefficient is  1 .
The middle term is,  +20x  its coefficient is  20 .
The last term, "the constant", is  -15000 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -15000 = -15000 

Step-2 : Find two factors of  -15000  whose sum equals the coefficient of the middle term, which is   20 .

For tidiness, printing of 34 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  :

  x2 + 20x - 15000  = 0 

Step  2  :

Parabola, Finding the Vertex :

 2.1      Find the Vertex of   y = x²+20x-15000

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,Ax²+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is  -10.0000  

 Plugging into the parabola formula  -10.0000  for  x  we can calculate the  y -coordinate : 
  y = 1.0 * -10.00 * -10.00 + 20.0 * -10.00 - 15000.0
or   y = -15100.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = x²+20x-15000
Axis of Symmetry (dashed)  {x}={-10.00} 
Vertex at  {x,y} = {-10.00,-15100.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-132.88, 0.00} 
Root 2 at  {x,y} = {112.88, 0.00} 

Solve Quadratic Equation by Completing The Square

 2.2     Solving   x²+20x-15000 = 0 by Completing The Square .

 Add  15000  to both side of the equation :
   x²+20x = 15000

Now the clever bit: Take the coefficient of  x , which is  20 , divide by two, giving  10 , and finally square it giving  100 

Add  100  to both sides of the equation :
  On the right hand side we have :
   15000  +  100    or,  (15000/1)+(100/1) 
  The common denominator of the two fractions is  1   Adding  (15000/1)+(100/1)  gives  15100/1 
  So adding to both sides we finally get :
   x²+20x+100 = 15100

Adding  100  has completed the left hand side into a perfect square :
   x²+20x+100  =
   (x+10) • (x+10)  =
  (x+10)²
Things which are equal to the same thing are also equal to one another. Since
   x²+20x+100 = 15100 and
   x²+20x+100 = (x+10)²
then, according to the law of transitivity,
   (x+10)² = 15100

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
   (x+10)²   is
   (x+10)^2/2 =
  (x+10)¹ =
   x+10

Now, applying the Square Root Principle to  Eq. #2.2.1  we get:
   x+10 = √ 15100

Subtract  10  from both sides to obtain:
   x = -10 + √ 15100

Since a square root has two values, one positive and the other negative
   x² + 20x - 15000 = 0
   has two solutions:
  x = -10 + √ 15100
   or
  x = -10 - √ 15100

Solve Quadratic Equation using the Quadratic Formula

 2.3     Solving    x²+20x-15000 = 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   =     1
                      B   =    20
                      C   =  -15000

Accordingly,  B²  -  4AC   =
                     400 - (-60000) =
                     60400

Applying the quadratic formula :

               -20 ± √ 60400
   x  =    ————————
                        2

Can  √ 60400 be simplified ?

Yes!   The prime factorization of  60400   is
   2•2•2•2•5•5•151 
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).

√ 60400   =  √ 2•2•2•2•5•5•151   =2•2•5•√ 151   =
                ±  20 • √ 151

  √ 151   , rounded to 4 decimal digits, is  12.2882
 So now we are looking at:
           x  =  ( -20 ± 20 •  12.288 ) / 2

Two real solutions:

 x =(-20+√60400)/2=-10+10√ 151 = 112.882

or:

 x =(-20-√60400)/2=-10-10√ 151 = -132.882

Two solutions were found :

1.  x =(-20-√60400)/2=-10-10√ 151 = -132.882
2.  x =(-20+√60400)/2=-10+10√ 151 = 112.882