Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((2•1669x2) +  5323x) -  77619  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  3338x²+5323x-77619 

The first term is,  3338x²  its coefficient is  3338 .
The middle term is,  +5323x  its coefficient is  5323 .
The last term, "the constant", is  -77619 

Step-1 : Multiply the coefficient of the first term by the constant   3338 • -77619 = -259092222 

Step-2 : Find two factors of  -259092222  whose sum equals the coefficient of the middle term, which is   5323 .


Numbers too big. Method shall not be applied

Equation at the end of step  2  :

  3338x2 + 5323x - 77619  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 3338x²+5323x-77619

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, 3338 , 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  -0.7973  

 Plugging into the parabola formula  -0.7973  for  x  we can calculate the  y -coordinate : 
  y = 3338.0 * -0.80 * -0.80 + 5323.0 * -0.80 - 77619.0
or   y = -79741.104

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 3338x²+5323x-77619
Axis of Symmetry (dashed)  {x}={-0.80} 
Vertex at  {x,y} = {-0.80,-79741.10} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-5.68, 0.00} 
Root 2 at  {x,y} = { 4.09, 0.00} 

Solve Quadratic Equation using the Quadratic Formula

 3.2     Solving    3338x²+5323x-77619 = 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   =     3338.00
     B   =    5323.00
     C   =    -77619.00

   B² = 28334329.00 
   4AC = -1036368888.00 
   B² - 4AC = 1064703217.00 
   SQRT(B²-4AC) =  32629.79
   x=( -5323.00 ± 32629.79) / 6676.00 
   x =  4.09029
   x =  -5.68496

Two solutions were found :

1.    x =  -5.68496
   
2.    x =  4.09029