Solution - Quadratic equations

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "24.01" was replaced by "(2401/100)".

Step by step solution :

Step  1  :

            2401
 Simplify   ————
            100 

Equation at the end of step  1  :

                  2401
  ((n2) +  2n) -  ————  = 0 
                  100 

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  100  as the denominator :

                n2 + 2n     (n2 + 2n) • 100
     n2 + 2n =  ———————  =  ———————————————
                   1              100      

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   n² + 2n  =   n • (n + 2) 

Adding fractions that have a common denominator :

 3.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

 n • (n+2) • 100 - (2401)     100n2 + 200n - 2401
 ————————————————————————  =  ———————————————————
           100                        100        

Trying to factor by splitting the middle term

 3.3     Factoring  100n² + 200n - 2401 

The first term is,  100n²  its coefficient is  100 .
The middle term is,  +200n  its coefficient is  200 .
The last term, "the constant", is  -2401 

Step-1 : Multiply the coefficient of the first term by the constant   100 • -2401 = -240100 

Step-2 : Find two factors of  -240100  whose sum equals the coefficient of the middle term, which is   200 .

For tidiness, printing of 39 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  3  :

  100n2 + 200n - 2401
  ———————————————————  = 0 
          100        

Step  4  :

When a fraction equals zero :

 4.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

  100n2+200n-2401
  ——————————————— • 100 = 0 • 100
        100      

Now, on the left hand side, the  100  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   100n²+200n-2401  = 0

Parabola, Finding the Vertex :

 4.2      Find the Vertex of   y = 100n²+200n-2401

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, 100 , 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,An²+Bn+C,the  n -coordinate of the vertex is given by  -B/(2A) . In our case the  n  coordinate is  -1.0000  

 Plugging into the parabola formula  -1.0000  for  n  we can calculate the  y -coordinate : 
  y = 100.0 * -1.00 * -1.00 + 200.0 * -1.00 - 2401.0
or   y = -2501.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 100n²+200n-2401
Axis of Symmetry (dashed)  {n}={-1.00} 
Vertex at  {n,y} = {-1.00,-2501.00} 
 n -Intercepts (Roots) :
Root 1 at  {n,y} = {-6.00, 0.00} 
Root 2 at  {n,y} = { 4.00, 0.00} 

Solve Quadratic Equation by Completing The Square

 4.3     Solving   100n²+200n-2401 = 0 by Completing The Square .

 Divide both sides of the equation by  100  to have 1 as the coefficient of the first term :
   n²+2n-(2401/100) = 0

Add  2401/100  to both side of the equation :
   n²+2n = 2401/100

Now the clever bit: Take the coefficient of  n , which is  2 , divide by two, giving  1 , and finally square it giving  1 

Add  1  to both sides of the equation :
  On the right hand side we have :
   2401/100  +  1    or,  (2401/100)+(1/1) 
  The common denominator of the two fractions is  100   Adding  (2401/100)+(100/100)  gives  2501/100 
  So adding to both sides we finally get :
   n²+2n+1 = 2501/100

Adding  1  has completed the left hand side into a perfect square :
   n²+2n+1  =
   (n+1) • (n+1)  =
  (n+1)²
Things which are equal to the same thing are also equal to one another. Since
   n²+2n+1 = 2501/100 and
   n²+2n+1 = (n+1)²
then, according to the law of transitivity,
   (n+1)² = 2501/100

We'll refer to this Equation as  Eq. #4.3.1  

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

Note that the square root of
   (n+1)²   is
   (n+1)^2/2 =
  (n+1)¹ =
   n+1

Now, applying the Square Root Principle to  Eq. #4.3.1  we get:
   n+1 = √ 2501/100

Subtract  1  from both sides to obtain:
   n = -1 + √ 2501/100

Since a square root has two values, one positive and the other negative
   n² + 2n - (2401/100) = 0
   has two solutions:
  n = -1 + √ 2501/100
   or
  n = -1 - √ 2501/100

Note that  √ 2501/100 can be written as
  √ 2501  / √ 100   which is √ 2501  / 10

Solve Quadratic Equation using the Quadratic Formula

 4.4     Solving    100n²+200n-2401 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  n  , the solution for   An²+Bn+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  n =   ————————
                      2A

  In our case,  A   =    100
                      B   =   200
                      C   =  -2401

Accordingly,  B²  -  4AC   =
                     40000 - (-960400) =
                     1000400

Applying the quadratic formula :

               -200 ± √ 1000400
   n  =    ——————————
                           200

Can  √ 1000400 be simplified ?

Yes!   The prime factorization of  1000400   is
   2•2•2•2•5•5•41•61 
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).

√ 1000400   =  √ 2•2•2•2•5•5•41•61   =2•2•5•√ 2501   =
                ±  20 • √ 2501

  √ 2501   , rounded to 4 decimal digits, is  50.0100
 So now we are looking at:
           n  =  ( -200 ± 20 •  50.010 ) / 200

Two real solutions:

 n =(-200+√1000400)/200=-1+1/10√ 2501 = 4.001

or:

 n =(-200-√1000400)/200=-1-1/10√ 2501 = -6.001

Two solutions were found :

1.  n =(-200-√1000400)/200=-1-1/10√ 2501 = -6.001
2.  n =(-200+√1000400)/200=-1+1/10√ 2501 = 4.001