Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((23•54x2) +  15x) -  16  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  5000x²+15x-16 

The first term is,  5000x²  its coefficient is  5000 .
The middle term is,  +15x  its coefficient is  15 .
The last term, "the constant", is  -16 

Step-1 : Multiply the coefficient of the first term by the constant   5000 • -16 = -80000 

Step-2 : Find two factors of  -80000  whose sum equals the coefficient of the middle term, which is   15 .

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  2  :

  5000x2 + 15x - 16  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 5000x²+15x-16

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, 5000 , 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  -0.0015  

 Plugging into the parabola formula  -0.0015  for  x  we can calculate the  y -coordinate : 
  y = 5000.0 * -0.00 * -0.00 + 15.0 * -0.00 - 16.0
or   y = -16.011

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 5000x²+15x-16
Axis of Symmetry (dashed)  {x}={-0.00} 
Vertex at  {x,y} = {-0.00,-16.01} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-0.06, 0.00} 
Root 2 at  {x,y} = { 0.06, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   5000x²+15x-16 = 0 by Completing The Square .

 Divide both sides of the equation by  5000  to have 1 as the coefficient of the first term :
   x²+(3/1000)x-(2/625) = 0

Add  2/625  to both side of the equation :
   x²+(3/1000)x = 2/625

Now the clever bit: Take the coefficient of  x , which is  3/1000 , divide by two, giving  3/2000 , and finally square it giving  3/2000 

Add  3/2000  to both sides of the equation :
  On the right hand side we have :
   2/625  +  3/2000   The common denominator of the two fractions is  10000   Adding  (32/10000)+(15/10000)  gives  47/10000 
  So adding to both sides we finally get :
   x²+(3/1000)x+(3/2000) = 47/10000

Adding  3/2000  has completed the left hand side into a perfect square :
   x²+(3/1000)x+(3/2000)  =
   (x+(3/2000)) • (x+(3/2000))  =
  (x+(3/2000))²
Things which are equal to the same thing are also equal to one another. Since
   x²+(3/1000)x+(3/2000) = 47/10000 and
   x²+(3/1000)x+(3/2000) = (x+(3/2000))²
then, according to the law of transitivity,
   (x+(3/2000))² = 47/10000

We'll refer to this Equation as  Eq. #3.2.1  

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

Note that the square root of
   (x+(3/2000))²   is
   (x+(3/2000))^2/2 =
  (x+(3/2000))¹ =
   x+(3/2000)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x+(3/2000) = √ 47/10000

Subtract  3/2000  from both sides to obtain:
   x = -3/2000 + √ 47/10000

Since a square root has two values, one positive and the other negative
   x² + (3/1000)x - (2/625) = 0
   has two solutions:
  x = -3/2000 + √ 47/10000
   or
  x = -3/2000 - √ 47/10000

Note that  √ 47/10000 can be written as
  √ 47  / √ 10000   which is √ 47  / 100

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    5000x²+15x-16 = 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   =    5000
                      B   =    15
                      C   =  -16

Accordingly,  B²  -  4AC   =
                     225 - (-320000) =
                     320225

Applying the quadratic formula :

               -15 ± √ 320225
   x  =    ————————
                        10000

Can  √ 320225 be simplified ?

Yes!   The prime factorization of  320225   is
   5•5•12809 
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).

√ 320225   =  √ 5•5•12809   =
                ±  5 • √ 12809

  √ 12809   , rounded to 4 decimal digits, is  113.1769
 So now we are looking at:
           x  =  ( -15 ± 5 •  113.177 ) / 10000

Two real solutions:

 x =(-15+√320225)/10000=(-3+√ 12809 )/2000= 0.055

or:

 x =(-15-√320225)/10000=(-3-√ 12809 )/2000= -0.058

Two solutions were found :

1.  x =(-15-√320225)/10000=(-3-√ 12809 )/2000= -0.058
2.  x =(-15+√320225)/10000=(-3+√ 12809 )/2000= 0.055