Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (7v2 -  53v) +  24  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  7v²-53v+24 

The first term is,  7v²  its coefficient is  7 .
The middle term is,  -53v  its coefficient is  -53 .
The last term, "the constant", is  +24 

Step-1 : Multiply the coefficient of the first term by the constant   7 • 24 = 168 

Step-2 : Find two factors of  168  whose sum equals the coefficient of the middle term, which is   -53 .

For tidiness, printing of 26 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  2  :

  7v2 - 53v + 24  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 7v²-53v+24

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, 7 , 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,Av²+Bv+C,the  v -coordinate of the vertex is given by  -B/(2A) . In our case the  v  coordinate is   3.7857  

 Plugging into the parabola formula   3.7857  for  v  we can calculate the  y -coordinate : 
  y = 7.0 * 3.79 * 3.79 - 53.0 * 3.79 + 24.0
or   y = -76.321

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 7v²-53v+24
Axis of Symmetry (dashed)  {v}={ 3.79} 
Vertex at  {v,y} = { 3.79,-76.32} 
 v -Intercepts (Roots) :
Root 1 at  {v,y} = { 0.48, 0.00} 
Root 2 at  {v,y} = { 7.09, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   7v²-53v+24 = 0 by Completing The Square .

 Divide both sides of the equation by  7  to have 1 as the coefficient of the first term :
   v²-(53/7)v+(24/7) = 0

Subtract  24/7  from both side of the equation :
   v²-(53/7)v = -24/7

Now the clever bit: Take the coefficient of  v , which is  53/7 , divide by two, giving  53/14 , and finally square it giving  2809/196 

Add  2809/196  to both sides of the equation :
  On the right hand side we have :
   -24/7  +  2809/196   The common denominator of the two fractions is  196   Adding  (-672/196)+(2809/196)  gives  2137/196 
  So adding to both sides we finally get :
   v²-(53/7)v+(2809/196) = 2137/196

Adding  2809/196  has completed the left hand side into a perfect square :
   v²-(53/7)v+(2809/196)  =
   (v-(53/14)) • (v-(53/14))  =
  (v-(53/14))²
Things which are equal to the same thing are also equal to one another. Since
   v²-(53/7)v+(2809/196) = 2137/196 and
   v²-(53/7)v+(2809/196) = (v-(53/14))²
then, according to the law of transitivity,
   (v-(53/14))² = 2137/196

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
   (v-(53/14))²   is
   (v-(53/14))^2/2 =
  (v-(53/14))¹ =
   v-(53/14)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   v-(53/14) = √ 2137/196

Add  53/14  to both sides to obtain:
   v = 53/14 + √ 2137/196

Since a square root has two values, one positive and the other negative
   v² - (53/7)v + (24/7) = 0
   has two solutions:
  v = 53/14 + √ 2137/196
   or
  v = 53/14 - √ 2137/196

Note that  √ 2137/196 can be written as
  √ 2137  / √ 196   which is √ 2137  / 14

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    7v²-53v+24 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  v  , the solution for   Av²+Bv+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  v =   ————————
                      2A

  In our case,  A   =     7
                      B   =   -53
                      C   =   24

Accordingly,  B²  -  4AC   =
                     2809 - 672 =
                     2137

Applying the quadratic formula :

               53 ± √ 2137
   v  =    ——————
                      14

  √ 2137   , rounded to 4 decimal digits, is  46.2277
 So now we are looking at:
           v  =  ( 53 ±  46.228 ) / 14

Two real solutions:

 v =(53+√2137)/14= 7.088

or:

 v =(53-√2137)/14= 0.484

Two solutions were found :

1.  v =(53-√2137)/14= 0.484
2.  v =(53+√2137)/14= 7.088