Solution - Quadratic equations

Rearrange:

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

                     26*x^2-24*x-(106)=0 

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  ((2•13x2) -  24x) -  106  = 0 

Step  2  :

Step  3  :

Pulling out like terms :

 3.1     Pull out like factors :

   26x² - 24x - 106  =   2 • (13x² - 12x - 53) 

Trying to factor by splitting the middle term

 3.2     Factoring  13x² - 12x - 53 

The first term is,  13x²  its coefficient is  13 .
The middle term is,  -12x  its coefficient is  -12 .
The last term, "the constant", is  -53 

Step-1 : Multiply the coefficient of the first term by the constant   13 • -53 = -689 

Step-2 : Find two factors of  -689  whose sum equals the coefficient of the middle term, which is   -12 .

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

Equation at the end of step  3  :

  2 • (13x2 - 12x - 53)  = 0 

Step  4  :

Equations which are never true :

 4.1      Solve :    2   =  0

This equation has no solution.
A a non-zero constant never equals zero.

Parabola, Finding the Vertex :

 4.2      Find the Vertex of   y = 13x²-12x-53

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, 13 , 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   0.4615  

 Plugging into the parabola formula   0.4615  for  x  we can calculate the  y -coordinate : 
  y = 13.0 * 0.46 * 0.46 - 12.0 * 0.46 - 53.0
or   y = -55.769

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 13x²-12x-53
Axis of Symmetry (dashed)  {x}={ 0.46} 
Vertex at  {x,y} = { 0.46,-55.77} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {-1.61, 0.00} 
Root 2 at  {x,y} = { 2.53, 0.00} 

Solve Quadratic Equation by Completing The Square

 4.3     Solving   13x²-12x-53 = 0 by Completing The Square .

 Divide both sides of the equation by  13  to have 1 as the coefficient of the first term :
   x²-(12/13)x-(53/13) = 0

Add  53/13  to both side of the equation :
   x²-(12/13)x = 53/13

Now the clever bit: Take the coefficient of  x , which is  12/13 , divide by two, giving  6/13 , and finally square it giving  36/169 

Add  36/169  to both sides of the equation :
  On the right hand side we have :
   53/13  +  36/169   The common denominator of the two fractions is  169   Adding  (689/169)+(36/169)  gives  725/169 
  So adding to both sides we finally get :
   x²-(12/13)x+(36/169) = 725/169

Adding  36/169  has completed the left hand side into a perfect square :
   x²-(12/13)x+(36/169)  =
   (x-(6/13)) • (x-(6/13))  =
  (x-(6/13))²
Things which are equal to the same thing are also equal to one another. Since
   x²-(12/13)x+(36/169) = 725/169 and
   x²-(12/13)x+(36/169) = (x-(6/13))²
then, according to the law of transitivity,
   (x-(6/13))² = 725/169

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
   (x-(6/13))²   is
   (x-(6/13))^2/2 =
  (x-(6/13))¹ =
   x-(6/13)

Now, applying the Square Root Principle to  Eq. #4.3.1  we get:
   x-(6/13) = √ 725/169

Add  6/13  to both sides to obtain:
   x = 6/13 + √ 725/169

Since a square root has two values, one positive and the other negative
   x² - (12/13)x - (53/13) = 0
   has two solutions:
  x = 6/13 + √ 725/169
   or
  x = 6/13 - √ 725/169

Note that  √ 725/169 can be written as
  √ 725  / √ 169   which is √ 725  / 13

Solve Quadratic Equation using the Quadratic Formula

 4.4     Solving    13x²-12x-53 = 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   =     13
                      B   =   -12
                      C   =  -53

Accordingly,  B²  -  4AC   =
                     144 - (-2756) =
                     2900

Applying the quadratic formula :

               12 ± √ 2900
   x  =    ——————
                      26

Can  √ 2900 be simplified ?

Yes!   The prime factorization of  2900   is
   2•2•5•5•29 
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).

√ 2900   =  √ 2•2•5•5•29   =2•5•√ 29   =
                ±  10 • √ 29

  √ 29   , rounded to 4 decimal digits, is   5.3852
 So now we are looking at:
           x  =  ( 12 ± 10 •  5.385 ) / 26

Two real solutions:

 x =(12+√2900)/26=(6+5√ 29 )/13= 2.533

or:

 x =(12-√2900)/26=(6-5√ 29 )/13= -1.610

Two solutions were found :

1.  x =(12-√2900)/26=(6-5√ 29 )/13= -1.610
2.  x =(12+√2900)/26=(6+5√ 29 )/13= 2.533