Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (2557x2 +  2752x) -  177658  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  2557x²+2752x-177658 

The first term is,  2557x²  its coefficient is  2557 .
The middle term is,  +2752x  its coefficient is  2752 .
The last term, "the constant", is  -177658 

Step-1 : Multiply the coefficient of the first term by the constant   2557 • -177658 = -454271506 

Step-2 : Find two factors of  -454271506  whose sum equals the coefficient of the middle term, which is   2752 .


Numbers too big. Method shall not be applied

Equation at the end of step  2  :

  2557x2 + 2752x - 177658  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 2557x²+2752x-177658

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, 2557 , 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  -0.5381  

 Plugging into the parabola formula  -0.5381  for  x  we can calculate the  y -coordinate : 
  y = 2557.0 * -0.54 * -0.54 + 2752.0 * -0.54 - 177658.0
or   y = -178398.468

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 2557x²+2752x-177658
Axis of Symmetry (dashed)  {x}={-0.54} 
Vertex at  {x,y} = {-0.54,-178398.47} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-8.89, 0.00} 
Root 2 at  {x,y} = { 7.81, 0.00} 

Solve Quadratic Equation using the Quadratic Formula

 3.2     Solving    2557x²+2752x-177658 = 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   =     2557.00
     B   =    2752.00
     C   =    -177658.00

   B² = 7573504.00 
   4AC = -1817086024.00 
   B² - 4AC = 1824659528.00 
   SQRT(B²-4AC) =  42716.03
   x=( -2752.00 ± 42716.03) / 5114.00 
   x =  7.81463
   x =  -8.89089

Two solutions were found :

1.    x =  -8.89089
   
2.    x =  7.81463