Solution - Quadratic equations

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "0.1025" was replaced by "(1025/10000)".

Step by step solution :

Step  1  :

             41
 Simplify   ———
            400

Equation at the end of step  1  :

                   41
  ((r2) +  2r) -  ———  = 0 
                  400

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  400  as the denominator :

                r2 + 2r     (r2 + 2r) • 400
     r2 + 2r =  ———————  =  ———————————————
                   1              400      

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 :

   r² + 2r  =   r • (r + 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:

 r • (r+2) • 400 - (41)     400r2 + 800r - 41
 ——————————————————————  =  —————————————————
          400                      400       

Trying to factor by splitting the middle term

 3.3     Factoring  400r² + 800r - 41 

The first term is,  400r²  its coefficient is  400 .
The middle term is,  +800r  its coefficient is  800 .
The last term, "the constant", is  -41 

Step-1 : Multiply the coefficient of the first term by the constant   400 • -41 = -16400 

Step-2 : Find two factors of  -16400  whose sum equals the coefficient of the middle term, which is   800 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -20  and  820 
                     400r² - 20r + 820r - 41

Step-4 : Add up the first 2 terms, pulling out like factors :
                    20r • (20r-1)
              Add up the last 2 terms, pulling out common factors :
                    41 • (20r-1)
Step-5 : Add up the four terms of step 4 :
                    (20r+41)  •  (20r-1)
             Which is the desired factorization

Equation at the end of step  3  :

  (20r - 1) • (20r + 41)
  ——————————————————————  = 0 
           400          

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:

  (20r-1)•(20r+41)
  ———————————————— • 400 = 0 • 400
        400       

Now, on the left hand side, the  400  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   (20r-1)  •  (20r+41)  = 0

Theory - Roots of a product :

 4.2    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.

Solving a Single Variable Equation :

 4.3      Solve  :    20r-1 = 0 

 Add  1  to both sides of the equation : 
                      20r = 1
Divide both sides of the equation by 20:
                     r = 1/20 = 0.050

Solving a Single Variable Equation :

 4.4      Solve  :    20r+41 = 0 

 Subtract  41  from both sides of the equation : 
                      20r = -41
Divide both sides of the equation by 20:
                     r = -41/20 = -2.050

Supplement : Solving Quadratic Equation Directly

Solving    400r2+800r-41  = 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 :

 5.1      Find the Vertex of   y = 400r²+800r-41

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, 400 , 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,Ar²+Br+C,the  r -coordinate of the vertex is given by  -B/(2A) . In our case the  r  coordinate is  -1.0000  

 Plugging into the parabola formula  -1.0000  for  r  we can calculate the  y -coordinate : 
  y = 400.0 * -1.00 * -1.00 + 800.0 * -1.00 - 41.0
or   y = -441.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 400r²+800r-41
Axis of Symmetry (dashed)  {r}={-1.00} 
Vertex at  {r,y} = {-1.00,-441.00} 
 r -Intercepts (Roots) :
Root 1 at  {r,y} = {-2.05, 0.00} 
Root 2 at  {r,y} = { 0.05, 0.00} 

Solve Quadratic Equation by Completing The Square

 5.2     Solving   400r²+800r-41 = 0 by Completing The Square .

 Divide both sides of the equation by  400  to have 1 as the coefficient of the first term :
   r²+2r-(41/400) = 0

Add  41/400  to both side of the equation :
   r²+2r = 41/400

Now the clever bit: Take the coefficient of  r , 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 :
   41/400  +  1    or,  (41/400)+(1/1) 
  The common denominator of the two fractions is  400   Adding  (41/400)+(400/400)  gives  441/400 
  So adding to both sides we finally get :
   r²+2r+1 = 441/400

Adding  1  has completed the left hand side into a perfect square :
   r²+2r+1  =
   (r+1) • (r+1)  =
  (r+1)²
Things which are equal to the same thing are also equal to one another. Since
   r²+2r+1 = 441/400 and
   r²+2r+1 = (r+1)²
then, according to the law of transitivity,
   (r+1)² = 441/400

We'll refer to this Equation as  Eq. #5.2.1  

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

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

Now, applying the Square Root Principle to  Eq. #5.2.1  we get:
   r+1 = √ 441/400

Subtract  1  from both sides to obtain:
   r = -1 + √ 441/400

Since a square root has two values, one positive and the other negative
   r² + 2r - (41/400) = 0
   has two solutions:
  r = -1 + √ 441/400
   or
  r = -1 - √ 441/400

Note that  √ 441/400 can be written as
  √ 441  / √ 400   which is 21 / 20

Solve Quadratic Equation using the Quadratic Formula

 5.3     Solving    400r²+800r-41 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  r  , the solution for   Ar²+Br+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  r =   ————————
                      2A

  In our case,  A   =    400
                      B   =   800
                      C   =  -41

Accordingly,  B²  -  4AC   =
                     640000 - (-65600) =
                     705600

Applying the quadratic formula :

               -800 ± √ 705600
   r  =    —————————
                          800

Can  √ 705600 be simplified ?

Yes!   The prime factorization of  705600   is
   2•2•2•2•2•2•3•3•5•5•7•7 
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).

√ 705600   =  √ 2•2•2•2•2•2•3•3•5•5•7•7   =2•2•2•3•5•7•√ 1   =
                ±  840 • √ 1   =
                ±  840

So now we are looking at:
           r  =  ( -800 ± 840) / 800

Two real solutions:

r =(-800+√705600)/800=-1+21/20= 0.050

or:

r =(-800-√705600)/800=-1-21/20= -2.050

Two solutions were found :

1.  r = -41/20 = -2.050
   
2.  r = 1/20 = 0.050