Solution - Quadratic equations

Rearrange:

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

                     x*(6*x+100)-(50000)=0 

Step by step solution :

Step  1  :

Step  2  :

Pulling out like terms :

 2.1     Pull out like factors :

   6x + 100  =   2 • (3x + 50) 

Equation at the end of step  2  :

  2x • (3x + 50) -  50000  = 0 

Step  3  :

Step  4  :

Pulling out like terms :

 4.1     Pull out like factors :

   6x² + 100x - 50000  =   2 • (3x² + 50x - 25000) 

Trying to factor by splitting the middle term

 4.2     Factoring  3x² + 50x - 25000 

The first term is,  3x²  its coefficient is  3 .
The middle term is,  +50x  its coefficient is  50 .
The last term, "the constant", is  -25000 

Step-1 : Multiply the coefficient of the first term by the constant   3 • -25000 = -75000 

Step-2 : Find two factors of  -75000  whose sum equals the coefficient of the middle term, which is   50 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -250  and  300 
                     3x² - 250x + 300x - 25000

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (3x-250)
              Add up the last 2 terms, pulling out common factors :
                    100 • (3x-250)
Step-5 : Add up the four terms of step 4 :
                    (x+100)  •  (3x-250)
             Which is the desired factorization

Equation at the end of step  4  :

  2 • (3x - 250) • (x + 100)  = 0 

Step  5  :

Theory - Roots of a product :

 5.1    A product of several terms equals zero. 

 When a product of two or more terms equals zero, then at least one of the terms must be zero. 

 We shall now solve each term = 0 separately 

 In other words, we are going to solve as many equations as there are terms in the product 

 Any solution of term = 0 solves product = 0 as well.

Equations which are never true :

 5.2      Solve :    2   =  0

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

Solving a Single Variable Equation :

 5.3      Solve  :    3x-250 = 0 

 Add  250  to both sides of the equation : 
                      3x = 250
Divide both sides of the equation by 3:
                     x = 250/3 = 83.333

Solving a Single Variable Equation :

 5.4      Solve  :    x+100 = 0 

 Subtract  100  from both sides of the equation : 
                      x = -100

Supplement : Solving Quadratic Equation Directly

Solving    3x2+50x-25000  = 0   directly 

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex :

 6.1      Find the Vertex of   y = 3x²+50x-25000

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, 3 , 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  -8.3333  

 Plugging into the parabola formula  -8.3333  for  x  we can calculate the  y -coordinate : 
  y = 3.0 * -8.33 * -8.33 + 50.0 * -8.33 - 25000.0
or   y = -25208.333

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 3x²+50x-25000
Axis of Symmetry (dashed)  {x}={-8.33} 
Vertex at  {x,y} = {-8.33,-25208.33} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-100.00, 0.00} 
Root 2 at  {x,y} = {83.33, 0.00} 

Solve Quadratic Equation by Completing The Square

 6.2     Solving   3x²+50x-25000 = 0 by Completing The Square .

 Divide both sides of the equation by  3  to have 1 as the coefficient of the first term :
   x²+(50/3)x-(25000/3) = 0

Add  25000/3  to both side of the equation :
   x²+(50/3)x = 25000/3

Now the clever bit: Take the coefficient of  x , which is  50/3 , divide by two, giving  25/3 , and finally square it giving  625/9 

Add  625/9  to both sides of the equation :
  On the right hand side we have :
   25000/3  +  625/9   The common denominator of the two fractions is  9   Adding  (75000/9)+(625/9)  gives  75625/9 
  So adding to both sides we finally get :
   x²+(50/3)x+(625/9) = 75625/9

Adding  625/9  has completed the left hand side into a perfect square :
   x²+(50/3)x+(625/9)  =
   (x+(25/3)) • (x+(25/3))  =
  (x+(25/3))²
Things which are equal to the same thing are also equal to one another. Since
   x²+(50/3)x+(625/9) = 75625/9 and
   x²+(50/3)x+(625/9) = (x+(25/3))²
then, according to the law of transitivity,
   (x+(25/3))² = 75625/9

We'll refer to this Equation as  Eq. #6.2.1  

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

Note that the square root of
   (x+(25/3))²   is
   (x+(25/3))^2/2 =
  (x+(25/3))¹ =
   x+(25/3)

Now, applying the Square Root Principle to  Eq. #6.2.1  we get:
   x+(25/3) = √ 75625/9

Subtract  25/3  from both sides to obtain:
   x = -25/3 + √ 75625/9

Since a square root has two values, one positive and the other negative
   x² + (50/3)x - (25000/3) = 0
   has two solutions:
  x = -25/3 + √ 75625/9
   or
  x = -25/3 - √ 75625/9

Note that  √ 75625/9 can be written as
  √ 75625  / √ 9   which is 275 / 3

Solve Quadratic Equation using the Quadratic Formula

 6.3     Solving    3x²+50x-25000 = 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   =     3
                      B   =    50
                      C   =  -25000

Accordingly,  B²  -  4AC   =
                     2500 - (-300000) =
                     302500

Applying the quadratic formula :

               -50 ± √ 302500
   x  =    ————————
                        6

Can  √ 302500 be simplified ?

Yes!   The prime factorization of  302500   is
   2•2•5•5•5•5•11•11 
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).

√ 302500   =  √ 2•2•5•5•5•5•11•11   =2•5•5•11•√ 1   =
                ±  550 • √ 1   =
                ±  550

So now we are looking at:
           x  =  ( -50 ± 550) / 6

Two real solutions:

x =(-50+√302500)/6=(-25+275)/3= 83.333

or:

x =(-50-√302500)/6=(-25-275)/3= -100.000

Two solutions were found :

1.  x = -100
2.  x = 250/3 = 83.333