Solution - Quadratic equations

Rearrange:

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

                     4*x^2-35*x-(49)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (22x2 -  35x) -  49  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  4x²-35x-49 

The first term is,  4x²  its coefficient is  4 .
The middle term is,  -35x  its coefficient is  -35 .
The last term, "the constant", is  -49 

Step-1 : Multiply the coefficient of the first term by the constant   4 • -49 = -196 

Step-2 : Find two factors of  -196  whose sum equals the coefficient of the middle term, which is   -35 .

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

Equation at the end of step  2  :

  4x2 - 35x - 49  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 4x²-35x-49

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, 4 , 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   4.3750  

 Plugging into the parabola formula   4.3750  for  x  we can calculate the  y -coordinate : 
  y = 4.0 * 4.38 * 4.38 - 35.0 * 4.38 - 49.0
or   y = -125.562

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 4x²-35x-49
Axis of Symmetry (dashed)  {x}={ 4.38} 
Vertex at  {x,y} = { 4.38,-125.56} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-1.23, 0.00} 
Root 2 at  {x,y} = { 9.98, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   4x²-35x-49 = 0 by Completing The Square .

 Divide both sides of the equation by  4  to have 1 as the coefficient of the first term :
   x²-(35/4)x-(49/4) = 0

Add  49/4  to both side of the equation :
   x²-(35/4)x = 49/4

Now the clever bit: Take the coefficient of  x , which is  35/4 , divide by two, giving  35/8 , and finally square it giving  1225/64 

Add  1225/64  to both sides of the equation :
  On the right hand side we have :
   49/4  +  1225/64   The common denominator of the two fractions is  64   Adding  (784/64)+(1225/64)  gives  2009/64 
  So adding to both sides we finally get :
   x²-(35/4)x+(1225/64) = 2009/64

Adding  1225/64  has completed the left hand side into a perfect square :
   x²-(35/4)x+(1225/64)  =
   (x-(35/8)) • (x-(35/8))  =
  (x-(35/8))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(35/4)x+(1225/64) = 2009/64 and
   x²-(35/4)x+(1225/64) = (x-(35/8))²
then, according to the law of transitivity,
   (x-(35/8))² = 2009/64

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-(35/8))²   is
   (x-(35/8))^2/2 =
  (x-(35/8))¹ =
   x-(35/8)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   x-(35/8) = √ 2009/64

Add  35/8  to both sides to obtain:
   x = 35/8 + √ 2009/64

Since a square root has two values, one positive and the other negative
   x² - (35/4)x - (49/4) = 0
   has two solutions:
  x = 35/8 + √ 2009/64
   or
  x = 35/8 - √ 2009/64

Note that  √ 2009/64 can be written as
  √ 2009  / √ 64   which is √ 2009  / 8

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    4x²-35x-49 = 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   =     4
                      B   =   -35
                      C   =  -49

Accordingly,  B²  -  4AC   =
                     1225 - (-784) =
                     2009

Applying the quadratic formula :

               35 ± √ 2009
   x  =    ——————
                      8

Can  √ 2009 be simplified ?

Yes!   The prime factorization of  2009   is
   7•7•41 
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).

√ 2009   =  √ 7•7•41   =
                ±  7 • √ 41

  √ 41   , rounded to 4 decimal digits, is   6.4031
 So now we are looking at:
           x  =  ( 35 ± 7 •  6.403 ) / 8

Two real solutions:

 x =(35+√2009)/8=(35+7√ 41 )/8= 9.978

or:

 x =(35-√2009)/8=(35-7√ 41 )/8= -1.228

Two solutions were found :

1.  x =(35-√2009)/8=(35-7√ 41 )/8= -1.228
2.  x =(35+√2009)/8=(35+7√ 41 )/8= 9.978