Solution - Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

  (32a2 +  25a) +  7  = 0 

Step  2  :

Trying to factor by splitting the middle term

 2.1     Factoring  9a²+25a+7 

The first term is,  9a²  its coefficient is  9 .
The middle term is,  +25a  its coefficient is  25 .
The last term, "the constant", is  +7 

Step-1 : Multiply the coefficient of the first term by the constant   9 • 7 = 63 

Step-2 : Find two factors of  63  whose sum equals the coefficient of the middle term, which is   25 .

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

Equation at the end of step  2  :

  9a2 + 25a + 7  = 0 

Step  3  :

Parabola, Finding the Vertex :

 3.1      Find the Vertex of   y = 9a²+25a+7

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, 9 , 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,Aa²+Ba+C,the  a -coordinate of the vertex is given by  -B/(2A) . In our case the  a  coordinate is  -1.3889  

 Plugging into the parabola formula  -1.3889  for  a  we can calculate the  y -coordinate : 
  y = 9.0 * -1.39 * -1.39 + 25.0 * -1.39 + 7.0
or   y = -10.361

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 9a²+25a+7
Axis of Symmetry (dashed)  {a}={-1.39} 
Vertex at  {a,y} = {-1.39,-10.36} 
 a -Intercepts (Roots) :
Root 1 at  {a,y} = {-2.46, 0.00} 
Root 2 at  {a,y} = {-0.32, 0.00} 

Solve Quadratic Equation by Completing The Square

 3.2     Solving   9a²+25a+7 = 0 by Completing The Square .

 Divide both sides of the equation by  9  to have 1 as the coefficient of the first term :
   a²+(25/9)a+(7/9) = 0

Subtract  7/9  from both side of the equation :
   a²+(25/9)a = -7/9

Now the clever bit: Take the coefficient of  a , which is  25/9 , divide by two, giving  25/18 , and finally square it giving  625/324 

Add  625/324  to both sides of the equation :
  On the right hand side we have :
   -7/9  +  625/324   The common denominator of the two fractions is  324   Adding  (-252/324)+(625/324)  gives  373/324 
  So adding to both sides we finally get :
   a²+(25/9)a+(625/324) = 373/324

Adding  625/324  has completed the left hand side into a perfect square :
   a²+(25/9)a+(625/324)  =
   (a+(25/18)) • (a+(25/18))  =
  (a+(25/18))²
Things which are equal to the same thing are also equal to one another. Since
   a²+(25/9)a+(625/324) = 373/324 and
   a²+(25/9)a+(625/324) = (a+(25/18))²
then, according to the law of transitivity,
   (a+(25/18))² = 373/324

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
   (a+(25/18))²   is
   (a+(25/18))^2/2 =
  (a+(25/18))¹ =
   a+(25/18)

Now, applying the Square Root Principle to  Eq. #3.2.1  we get:
   a+(25/18) = √ 373/324

Subtract  25/18  from both sides to obtain:
   a = -25/18 + √ 373/324

Since a square root has two values, one positive and the other negative
   a² + (25/9)a + (7/9) = 0
   has two solutions:
  a = -25/18 + √ 373/324
   or
  a = -25/18 - √ 373/324

Note that  √ 373/324 can be written as
  √ 373  / √ 324   which is √ 373  / 18

Solve Quadratic Equation using the Quadratic Formula

 3.3     Solving    9a²+25a+7 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  a  , the solution for   Aa²+Ba+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  a =   ————————
                      2A

  In our case,  A   =     9
                      B   =    25
                      C   =   7

Accordingly,  B²  -  4AC   =
                     625 - 252 =
                     373

Applying the quadratic formula :

               -25 ± √ 373
   a  =    ——————
                      18

  √ 373   , rounded to 4 decimal digits, is  19.3132
 So now we are looking at:
           a  =  ( -25 ±  19.313 ) / 18

Two real solutions:

 a =(-25+√373)/18=-0.316

or:

 a =(-25-√373)/18=-2.462

Two solutions were found :

1.  a =(-25-√373)/18=-2.462
2.  a =(-25+√373)/18=-0.316