Solution - Linear inequalities

Step by step solution :

Step  1  :

Trying to factor by splitting the middle term

 1.1     Factoring  x²-100x+2500 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -100x  its coefficient is  -100 .
The last term, "the constant", is  +2500 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 2500 = 2500 

Step-2 : Find two factors of  2500  whose sum equals the coefficient of the middle term, which is   -100 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -50  and  -50 
                     x² - 50x - 50x - 2500

Step-4 : Add up the first 2 terms, pulling out like factors :
                    x • (x-50)
              Add up the last 2 terms, pulling out common factors :
                    50 • (x-50)
Step-5 : Add up the four terms of step 4 :
                    (x-50)  •  (x-50)
             Which is the desired factorization

Multiplying Exponential Expressions :

 1.2    Multiply  (x-50)  by  (x-50) 

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is  (x-50)  and the exponents are :
          1 , as  (x-50)  is the same number as  (x-50)¹ 
 and   1 , as  (x-50)  is the same number as  (x-50)¹ 
The product is therefore,  (x-50)⁽¹⁺¹⁾ = (x-50)² 

Equation at the end of step  1  :

  (x - 50)2  ≥ 0 

Step  2  :

Solve Basic Inequality :

 2.1      Add  50  to both sides

             (x)2  ≥ 50

Inequality Plot :

 2.2      Inequality plot for

         x  - 50.000  ≥  0

  
 

Supplement : Solving Quadratic Equation Directly

Solving    x2-100x+2500  = 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 :

 3.1      Find the Vertex of   y = x²-100x+2500

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  50.0000  

 Plugging into the parabola formula  50.0000  for  x  we can calculate the  y -coordinate : 
  y = 1.0 * 50.00 * 50.00 - 100.0 * 50.00 + 2500.0
or   y = 0.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = x²-100x+2500
Vertex at  {x,y} = {50.00, 0.00} 
x-Intercept (Root) :
One Root at  {x,y}={50.00, 0.00} 
Note that the root coincides with
the Vertex and the Axis of Symmetry
coinsides with the line  x = 0 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   x²-100x+2500 = 0 by Completing The Square .

 Subtract  2500  from both side of the equation :
   x²-100x = -2500

Now the clever bit: Take the coefficient of  x , which is  100 , divide by two, giving  50 , and finally square it giving  2500 

Add  2500  to both sides of the equation :
  On the right hand side we have :
   -2500  +  2500    or,  (-2500/1)+(2500/1) 
  The common denominator of the two fractions is  1   Adding  (-2500/1)+(2500/1)  gives  0/1 
  So adding to both sides we finally get :
   x²-100x+2500 = 0

Adding  2500  has completed the left hand side into a perfect square :
   x²-100x+2500  =
   (x-50) • (x-50)  =
  (x-50)²
Things which are equal to the same thing are also equal to one another. Since
   x²-100x+2500 = 0 and
   x²-100x+2500 = (x-50)²
then, according to the law of transitivity,
   (x-50)² = 0

We'll refer to this Equation as  Eq. #3.2.1  

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

Note that the square root of
   (x-50)²   is
   (x-50)^2/2 =
  (x-50)¹ =
   x-50

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x-50 = √ 0

Add  50  to both sides to obtain:
   x = 50 + √ 0
The square root of zero is zero

This quadratic equation has one solution only. That's because adding zero is the same as subtracting zero.

The solution is:
   x  =  50 

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    x²-100x+2500 = 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   =     1
                      B   =   -100
                      C   =  2500

Accordingly,  B²  -  4AC   =
                     10000 - 10000 =
                     0

Applying the quadratic formula :

               100 ± √ 0
   x  =    —————
                    2

The square root of zero is zero

This quadratic equation has one solution only. That's because adding zero is the same as subtracting zero.

The solution is:
  x = 100 / 2 = 50

One solution was found :