Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (53x2 +  785x) -  353250  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   125x² + 785x - 353250  =   5 • (25x² + 157x - 70650) 

Trying to factor by splitting the middle term

 3.2     Factoring  25x² + 157x - 70650 

The first term is,  25x²  its coefficient is  25 .
The middle term is,  +157x  its coefficient is  157 .
The last term, "the constant", is  -70650 

Step-1 : Multiply the coefficient of the first term by the constant   25 • -70650 = -1766250 

Step-2 : Find two factors of  -1766250  whose sum equals the coefficient of the middle term, which is   157 .


Numbers too big. Method shall not be applied

Equation at the end of step  3  :

  5 • (25x2 + 157x - 70650)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    5   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Parabola, Finding the Vertex :

 4.2      Find the Vertex of   y = 25x²+157x-70650

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, 25 , 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  -3.1400  

 Plugging into the parabola formula  -3.1400  for  x  we can calculate the  y -coordinate : 
  y = 25.0 * -3.14 * -3.14 + 157.0 * -3.14 - 70650.0
or   y = -70896.490

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 25x²+157x-70650
Axis of Symmetry (dashed)  {x}={-3.14} 
Vertex at  {x,y} = {-3.14,-70896.49} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-56.39, 0.00} 
Root 2 at  {x,y} = {50.11, 0.00} 

Solve Quadratic Equation using the Quadratic Formula

 4.3     Solving    25x²+157x-70650 = 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   =     25.00
     B   =    157.00
     C   =    -70650.00

   B² = 24649.00 
   4AC = -7065000.00 
   B² - 4AC = 7089649.00 
   SQRT(B²-4AC) =  2662.64
   x=( -157.00 ± 2662.64) / 50.00 
   x =  50.11279
   x =  -56.39279

Two solutions were found :

1.    x =  -56.39279
   
2.    x =  50.11279