Solution - Quadratic equations

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "0.5" was replaced by "(5/10)".

Step by step solution :

Step  1  :

            1
 Simplify   —
            2

Equation at the end of step  1  :

          1                    
  ((0 -  (— • x2)) +  25x) -  177  = 0 
          2                    

Step  2  :

Equation at the end of step  2  :

         x2             
  ((0 -  ——) +  25x) -  177  = 0 
         2              

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Adding a whole to a fraction

Rewrite the whole as a fraction using  2  as the denominator :

           25x     25x • 2
    25x =  ———  =  ———————
            1         2   

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

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:

 -x2 + 25x • 2     50x - x2
 —————————————  =  ————————
       2              2    

Equation at the end of step  3  :

  (50x - x2)    
  —————————— -  177  = 0 
      2         

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  2  as the denominator :

           177     177 • 2
    177 =  ———  =  ———————
            1         2   

Step  5  :

Pulling out like terms :

 5.1     Pull out like factors :

   50x - x²  =   -x • (x - 50) 

Adding fractions that have a common denominator :

 5.2       Adding up the two equivalent fractions

 -x • (x-50) - (177 • 2)     -x2 + 50x - 354
 ———————————————————————  =  ———————————————
            2                       2       

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   -x² + 50x - 354  =   -1 • (x² - 50x + 354) 

Trying to factor by splitting the middle term

 6.2     Factoring  x² - 50x + 354 

The first term is,  x²  its coefficient is  1 .
The middle term is,  -50x  its coefficient is  -50 .
The last term, "the constant", is  +354 

Step-1 : Multiply the coefficient of the first term by the constant   1 • 354 = 354 

Step-2 : Find two factors of  354  whose sum equals the coefficient of the middle term, which is   -50 .

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  6  :

  -x2 + 50x - 354
  ———————————————  = 0 
         2       

Step  7  :

When a fraction equals zero :

 7.1    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:

  -x2+50x-354
  ——————————— • 2 = 0 • 2
       2     

Now, on the left hand side, the  2  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   -x²+50x-354  = 0

Parabola, Finding the Vertex :

 7.2      Find the Vertex of   y = -x²+50x-354

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens down and accordingly has a highest point (AKA absolute maximum) .    We know this even before plotting  "y"  because the coefficient of the first term, -1 , is negative (smaller 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  25.0000  

 Plugging into the parabola formula  25.0000  for  x  we can calculate the  y -coordinate : 
  y = -1.0 * 25.00 * 25.00 + 50.0 * 25.00 - 354.0
or   y = 271.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = -x²+50x-354
Axis of Symmetry (dashed)  {x}={25.00} 
Vertex at  {x,y} = {25.00,271.00} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {41.46, 0.00} 
Root 2 at  {x,y} = { 8.54, 0.00} 

Solve Quadratic Equation by Completing The Square

 7.3     Solving   -x²+50x-354 = 0 by Completing The Square .

 Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term:
 x²-50x+354 = 0  Subtract  354  from both side of the equation :
   x²-50x = -354

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

Add  625  to both sides of the equation :
  On the right hand side we have :
   -354  +  625    or,  (-354/1)+(625/1) 
  The common denominator of the two fractions is  1   Adding  (-354/1)+(625/1)  gives  271/1 
  So adding to both sides we finally get :
   x²-50x+625 = 271

Adding  625  has completed the left hand side into a perfect square :
   x²-50x+625  =
   (x-25) • (x-25)  =
  (x-25)²
Things which are equal to the same thing are also equal to one another. Since
   x²-50x+625 = 271 and
   x²-50x+625 = (x-25)²
then, according to the law of transitivity,
   (x-25)² = 271

We'll refer to this Equation as  Eq. #7.3.1  

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

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

Now, applying the Square Root Principle to  Eq. #7.3.1  we get:
   x-25 = √ 271

Add  25  to both sides to obtain:
   x = 25 + √ 271

Since a square root has two values, one positive and the other negative
   x² - 50x + 354 = 0
   has two solutions:
  x = 25 + √ 271
   or
  x = 25 - √ 271

Solve Quadratic Equation using the Quadratic Formula

 7.4     Solving    -x²+50x-354 = 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   =    50
                      C   =  -354

Accordingly,  B²  -  4AC   =
                     2500 - 1416 =
                     1084

Applying the quadratic formula :

               -50 ± √ 1084
   x  =    ———————
                        -2

Can  √ 1084 be simplified ?

Yes!   The prime factorization of  1084   is
   2•2•271 
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).

√ 1084   =  √ 2•2•271   =
                ±  2 • √ 271

  √ 271   , rounded to 4 decimal digits, is  16.4621
 So now we are looking at:
           x  =  ( -50 ± 2 •  16.462 ) / -2

Two real solutions:

 x =(-50+√1084)/-2=25-√ 271 = 8.538

or:

 x =(-50-√1084)/-2=25+√ 271 = 41.462

Two solutions were found :

1.  x =(-50-√1084)/-2=25+√ 271 = 41.462
2.  x =(-50+√1084)/-2=25-√ 271 = 8.538