Solution - Quadratic equations

Step by step solution :

Step  1  :

            12
 Simplify   ——
            7 

Equation at the end of step  1  :

    10        2     12
  ((——•(z2))+(—•z))-——  = 0 
    5         2     7 

Step  2  :

            1
 Simplify   —
            1

Equation at the end of step  2  :

    10              12
  ((——•(z2))+(1•z))-——  = 0 
    5               7 
 Step  3  :
            2
 Simplify   —
            1

Equation at the end of step  3  :

                     12
  ((2 • z2) +  z) -  ——  = 0 
                     7 

Step  4  :

Equation at the end of step  4  :

                12
  (2z2 +  z) -  ——  = 0 
                7 

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  7  as the denominator :

                2z2 + z     (2z2 + z) • 7
     2z2 + z =  ———————  =  —————————————
                   1              7      

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

Step  6  :

Pulling out like terms :

 6.1     Pull out like factors :

   2z² + z  =   z • (2z + 1) 

Adding fractions that have a common denominator :

 6.2       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:

 z • (2z+1) • 7 - (12)     14z2 + 7z - 12
 —————————————————————  =  ——————————————
           7                     7       

Trying to factor by splitting the middle term

 6.3     Factoring  14z² + 7z - 12 

The first term is,  14z²  its coefficient is  14 .
The middle term is,  +7z  its coefficient is  7 .
The last term, "the constant", is  -12 

Step-1 : Multiply the coefficient of the first term by the constant   14 • -12 = -168 

Step-2 : Find two factors of  -168  whose sum equals the coefficient of the middle term, which is   7 .

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

Equation at the end of step  6  :

  14z2 + 7z - 12
  ——————————————  = 0 
        7       

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:

  14z2+7z-12
  —————————— • 7 = 0 • 7
      7     

Now, on the left hand side, the  7  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   14z²+7z-12  = 0

Parabola, Finding the Vertex :

 7.2      Find the Vertex of   y = 14z²+7z-12

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, 14 , 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,Az²+Bz+C,the  z -coordinate of the vertex is given by  -B/(2A) . In our case the  z  coordinate is  -0.2500  

 Plugging into the parabola formula  -0.2500  for  z  we can calculate the  y -coordinate : 
  y = 14.0 * -0.25 * -0.25 + 7.0 * -0.25 - 12.0
or   y = -12.875

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 14z²+7z-12
Axis of Symmetry (dashed)  {z}={-0.25} 
Vertex at  {z,y} = {-0.25,-12.88} 
 z -Intercepts (Roots) :
Root 1 at  {z,y} = {-1.21, 0.00} 
Root 2 at  {z,y} = { 0.71, 0.00} 

Solve Quadratic Equation by Completing The Square

 7.3     Solving   14z²+7z-12 = 0 by Completing The Square .

 Divide both sides of the equation by  14  to have 1 as the coefficient of the first term :
   z²+(1/2)z-(6/7) = 0

Add  6/7  to both side of the equation :
   z²+(1/2)z = 6/7

Now the clever bit: Take the coefficient of  z , which is  1/2 , divide by two, giving  1/4 , and finally square it giving  1/16 

Add  1/16  to both sides of the equation :
  On the right hand side we have :
   6/7  +  1/16   The common denominator of the two fractions is  112   Adding  (96/112)+(7/112)  gives  103/112 
  So adding to both sides we finally get :
   z²+(1/2)z+(1/16) = 103/112

Adding  1/16  has completed the left hand side into a perfect square :
   z²+(1/2)z+(1/16)  =
   (z+(1/4)) • (z+(1/4))  =
  (z+(1/4))²
Things which are equal to the same thing are also equal to one another. Since
   z²+(1/2)z+(1/16) = 103/112 and
   z²+(1/2)z+(1/16) = (z+(1/4))²
then, according to the law of transitivity,
   (z+(1/4))² = 103/112

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
   (z+(1/4))²   is
   (z+(1/4))^2/2 =
  (z+(1/4))¹ =
   z+(1/4)

Now, applying the Square Root Principle to  Eq. #7.3.1  we get:
   z+(1/4) = √ 103/112

Subtract  1/4  from both sides to obtain:
   z = -1/4 + √ 103/112

Since a square root has two values, one positive and the other negative
   z² + (1/2)z - (6/7) = 0
   has two solutions:
  z = -1/4 + √ 103/112
   or
  z = -1/4 - √ 103/112

Note that  √ 103/112 can be written as
  √ 103  / √ 112 

Solve Quadratic Equation using the Quadratic Formula

 7.4     Solving    14z²+7z-12 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  z  , the solution for   Az²+Bz+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  z =   ————————
                      2A

  In our case,  A   =     14
                      B   =    7
                      C   =  -12

Accordingly,  B²  -  4AC   =
                     49 - (-672) =
                     721

Applying the quadratic formula :

               -7 ± √ 721
   z  =    ——————
                      28

  √ 721   , rounded to 4 decimal digits, is  26.8514
 So now we are looking at:
           z  =  ( -7 ±  26.851 ) / 28

Two real solutions:

 z =(-7+√721)/28=-1/4+1/28√ 721 = 0.709

or:

 z =(-7-√721)/28=-1/4-1/28√ 721 = -1.209

Two solutions were found :

1.  z =(-7-√721)/28=-1/4-1/28√ 721 = -1.209
2.  z =(-7+√721)/28=-1/4+1/28√ 721 = 0.709