Solution - Quadratic equations

Rearrange:

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

                     40*x^2+94*x-39-(0/o)=0 

Step by step solution :

Step  1  :

            0
 Simplify   —
            o

Equation at the end of step  1  :

  (((40 • (x2)) +  94x) -  39) -  0  = 0 
 Step  2  :

Equation at the end of step  2  :

  (((23•5x2) +  94x) -  39) -  0  = 0 

Step  3  :

Trying to factor by splitting the middle term

 3.1     Factoring  40x²+94x-39 

The first term is,  40x²  its coefficient is  40 .
The middle term is,  +94x  its coefficient is  94 .
The last term, "the constant", is  -39 

Step-1 : Multiply the coefficient of the first term by the constant   40 • -39 = -1560 

Step-2 : Find two factors of  -1560  whose sum equals the coefficient of the middle term, which is   94 .

For tidiness, printing of 26 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  :

  40x2 + 94x - 39  = 0 

Step  4  :

Parabola, Finding the Vertex :

 4.1      Find the Vertex of   y = 40x²+94x-39

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, 40 , 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  -1.1750  

 Plugging into the parabola formula  -1.1750  for  x  we can calculate the  y -coordinate : 
  y = 40.0 * -1.18 * -1.18 + 94.0 * -1.18 - 39.0
or   y = -94.225

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 40x²+94x-39
Axis of Symmetry (dashed)  {x}={-1.18} 
Vertex at  {x,y} = {-1.18,-94.22} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-2.71, 0.00} 
Root 2 at  {x,y} = { 0.36, 0.00} 

Solve Quadratic Equation by Completing The Square

 4.2     Solving   40x²+94x-39 = 0 by Completing The Square .

 Divide both sides of the equation by  40  to have 1 as the coefficient of the first term :
   x²+(47/20)x-(39/40) = 0

Add  39/40  to both side of the equation :
   x²+(47/20)x = 39/40

Now the clever bit: Take the coefficient of  x , which is  47/20 , divide by two, giving  47/40 , and finally square it giving  2209/1600 

Add  2209/1600  to both sides of the equation :
  On the right hand side we have :
   39/40  +  2209/1600   The common denominator of the two fractions is  1600   Adding  (1560/1600)+(2209/1600)  gives  3769/1600 
  So adding to both sides we finally get :
   x²+(47/20)x+(2209/1600) = 3769/1600

Adding  2209/1600  has completed the left hand side into a perfect square :
   x²+(47/20)x+(2209/1600)  =
   (x+(47/40)) • (x+(47/40))  =
  (x+(47/40))²
Things which are equal to the same thing are also equal to one another. Since
   x²+(47/20)x+(2209/1600) = 3769/1600 and
   x²+(47/20)x+(2209/1600) = (x+(47/40))²
then, according to the law of transitivity,
   (x+(47/40))² = 3769/1600

We'll refer to this Equation as  Eq. #4.2.1  

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

Note that the square root of
   (x+(47/40))²   is
   (x+(47/40))^2/2 =
  (x+(47/40))¹ =
   x+(47/40)

Now, applying the Square Root Principle to  Eq. #4.2.1  we get:
   x+(47/40) = √ 3769/1600

Subtract  47/40  from both sides to obtain:
   x = -47/40 + √ 3769/1600

Since a square root has two values, one positive and the other negative
   x² + (47/20)x - (39/40) = 0
   has two solutions:
  x = -47/40 + √ 3769/1600
   or
  x = -47/40 - √ 3769/1600

Note that  √ 3769/1600 can be written as
  √ 3769  / √ 1600   which is √ 3769  / 40

Solve Quadratic Equation using the Quadratic Formula

 4.3     Solving    40x²+94x-39 = 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   =     40
                      B   =    94
                      C   =  -39

Accordingly,  B²  -  4AC   =
                     8836 - (-6240) =
                     15076

Applying the quadratic formula :

               -94 ± √ 15076
   x  =    ————————
                        80

Can  √ 15076 be simplified ?

Yes!   The prime factorization of  15076   is
   2•2•3769 
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).

√ 15076   =  √ 2•2•3769   =
                ±  2 • √ 3769

  √ 3769   , rounded to 4 decimal digits, is  61.3922
 So now we are looking at:
           x  =  ( -94 ± 2 •  61.392 ) / 80

Two real solutions:

 x =(-94+√15076)/80=(-47+√ 3769 )/40= 0.360

or:

 x =(-94-√15076)/80=(-47-√ 3769 )/40= -2.710

Two solutions were found :

1.  x =(-94-√15076)/80=(-47-√ 3769 )/40= -2.710
2.  x =(-94+√15076)/80=(-47+√ 3769 )/40= 0.360