Question

factorize x^2 + 250x + 750000

Answer 1

   Factoring  x2+250x-75000 

The first term is,  x2  its coefficient is  1 .
The middle term is,  +250x  its coefficient is  250 .
The last term, "the constant", is  -75000 

Step-1 : Multiply the coefficient of the first term by the constant   1 • -75000 = -75000 

Step-2 : Find two factors of  -75000  whose sum equals the coefficient of the middle term, which is   250 .

     -75000   +   1   =   -74999     -37500   +   2   =   -37498     -25000   +   3   =   -24997     -18750   +   4   =   -18746     -15000   +   5   =   -14995     -12500   +   6   =   -12494


For tidiness, printing of 42 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  1  :

x2 + 250x - 75000 = 0

Step  2  :

Parabola, Finding the Vertex :

 2.1      Find the Vertex of   y = x2+250x-75000

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, 1 , 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  -125.0000  

 Plugging into the parabola formula  -125.0000 for  x  we can calculate the  y -coordinate : 
  y = 1.0 * -125.00 * -125.00 + 250.0 * -125.00 - 75000.0 
or   y = -90625.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = x2+250x-75000
Axis of Symmetry (dashed)  {x}={-125.00} 
Vertex at  {x,y} = {-125.00,-90625.00}  
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-426.04, 0.00} 
Root 2 at  {x,y} = {1