Solution - Quadratic equations

Rearrange:

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

                     p^2/p-3-7/p-3-(9)=0 

Step by step solution :

Step  1  :

            7
 Simplify   —
            p

Equation at the end of step  1  :

     (p2)    7
  (((————-3)-—)-3)-9  = 0 
      p      p
 Step  2  :
            p2
 Simplify   ——
            p 

Dividing exponential expressions :

 2.1    p² divided by p¹ = p^{(2 - 1)} = p¹ = p

Equation at the end of step  2  :

                7           
  (((p -  3) -  —) -  3) -  9  = 0 
                p           

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

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

              p - 3     (p - 3) • p
     p - 3 =  —————  =  ———————————
                1            p     

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

Adding fractions that have a common denominator :

 3.2       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:

 (p-3) • p - (7)     p2 - 3p - 7
 ———————————————  =  ———————————
        p                 p     

Equation at the end of step  3  :

   (p2 - 3p - 7)          
  (————————————— -  3) -  9  = 0 
         p                

Step  4  :

Rewriting the whole as an Equivalent Fraction :

 4.1   Subtracting a whole from a fraction

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

         3     3 • p
    3 =  —  =  —————
         1       p  

Trying to factor by splitting the middle term

 4.2     Factoring  p² - 3p - 7 

The first term is,  p²  its coefficient is  1 .
The middle term is,  -3p  its coefficient is  -3 .
The last term, "the constant", is  -7 

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

Step-2 : Find two factors of  -7  whose sum equals the coefficient of the middle term, which is   -3 .

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

Adding fractions that have a common denominator :

 4.3       Adding up the two equivalent fractions

 (p2-3p-7) - (3 • p)     p2 - 6p - 7
 ———————————————————  =  ———————————
          p                   p     

Equation at the end of step  4  :

  (p2 - 6p - 7)    
  ————————————— -  9  = 0 
        p          

Step  5  :

Rewriting the whole as an Equivalent Fraction :

 5.1   Subtracting a whole from a fraction

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

         9     9 • p
    9 =  —  =  —————
         1       p  

Trying to factor by splitting the middle term

 5.2     Factoring  p² - 6p - 7 

The first term is,  p²  its coefficient is  1 .
The middle term is,  -6p  its coefficient is  -6 .
The last term, "the constant", is  -7 

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

Step-2 : Find two factors of  -7  whose sum equals the coefficient of the middle term, which is   -6 .

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -7  and  1 
                     p² - 7p + 1p - 7

Step-4 : Add up the first 2 terms, pulling out like factors :
                    p • (p-7)
              Add up the last 2 terms, pulling out common factors :
                     1 • (p-7)
Step-5 : Add up the four terms of step 4 :
                    (p+1)  •  (p-7)
             Which is the desired factorization

Adding fractions that have a common denominator :

 5.3       Adding up the two equivalent fractions

 (p+1) • (p-7) - (9 • p)     p2 - 15p - 7
 ———————————————————————  =  ————————————
            p                     p      

Trying to factor by splitting the middle term

 5.4     Factoring  p² - 15p - 7 

The first term is,  p²  its coefficient is  1 .
The middle term is,  -15p  its coefficient is  -15 .
The last term, "the constant", is  -7 

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

Step-2 : Find two factors of  -7  whose sum equals the coefficient of the middle term, which is   -15 .

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

Equation at the end of step  5  :

  p2 - 15p - 7
  ————————————  = 0 
       p      

Step  6  :

When a fraction equals zero :

 6.1    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:

  p2-15p-7
  ———————— • p = 0 • p
     p    

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

The equation now takes the shape :
   p²-15p-7  = 0

Parabola, Finding the Vertex :

 6.2      Find the Vertex of   y = p²-15p-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, 1 , 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,Ap²+Bp+C,the  p -coordinate of the vertex is given by  -B/(2A) . In our case the  p  coordinate is   7.5000  

 Plugging into the parabola formula   7.5000  for  p  we can calculate the  y -coordinate : 
  y = 1.0 * 7.50 * 7.50 - 15.0 * 7.50 - 7.0
or   y = -63.250

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = p²-15p-7
Axis of Symmetry (dashed)  {p}={ 7.50} 
Vertex at  {p,y} = { 7.50,-63.25} 
 p -Intercepts (Roots) :
Root 1 at  {p,y} = {-0.45, 0.00} 
Root 2 at  {p,y} = {15.45, 0.00} 

Solve Quadratic Equation by Completing The Square

 6.3     Solving   p²-15p-7 = 0 by Completing The Square .

 Add  7  to both side of the equation :
   p²-15p = 7

Now the clever bit: Take the coefficient of  p , which is  15 , divide by two, giving  15/2 , and finally square it giving  225/4 

Add  225/4  to both sides of the equation :
  On the right hand side we have :
   7  +  225/4    or,  (7/1)+(225/4) 
  The common denominator of the two fractions is  4   Adding  (28/4)+(225/4)  gives  253/4 
  So adding to both sides we finally get :
   p²-15p+(225/4) = 253/4

Adding  225/4  has completed the left hand side into a perfect square :
   p²-15p+(225/4)  =
   (p-(15/2)) • (p-(15/2))  =
  (p-(15/2))²
Things which are equal to the same thing are also equal to one another. Since
   p²-15p+(225/4) = 253/4 and
   p²-15p+(225/4) = (p-(15/2))²
then, according to the law of transitivity,
   (p-(15/2))² = 253/4

We'll refer to this Equation as  Eq. #6.3.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (p-(15/2))²   is
   (p-(15/2))^2/2 =
  (p-(15/2))¹ =
   p-(15/2)

Now, applying the Square Root Principle to  Eq. #6.3.1  we get:
   p-(15/2) = √ 253/4

Add  15/2  to both sides to obtain:
   p = 15/2 + √ 253/4

Since a square root has two values, one positive and the other negative
   p² - 15p - 7 = 0
   has two solutions:
  p = 15/2 + √ 253/4
   or
  p = 15/2 - √ 253/4

Note that  √ 253/4 can be written as
  √ 253  / √ 4   which is √ 253  / 2

Solve Quadratic Equation using the Quadratic Formula

 6.4     Solving    p²-15p-7 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  p  , the solution for   Ap²+Bp+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  p =   ————————
                      2A

  In our case,  A   =     1
                      B   =   -15
                      C   =   -7

Accordingly,  B²  -  4AC   =
                     225 - (-28) =
                     253

Applying the quadratic formula :

               15 ± √ 253
   p  =    ——————
                      2

  √ 253   , rounded to 4 decimal digits, is  15.9060
 So now we are looking at:
           p  =  ( 15 ±  15.906 ) / 2

Two real solutions:

 p =(15+√253)/2=15.453

or:

 p =(15-√253)/2=-0.453

Supplement : Solving Quadratic Equation Directly

Solving    p2-6p-7  = 0   directly 

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex :

 7.1      Find the Vertex of   y = p²-6p-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, 1 , 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,Ap²+Bp+C,the  p -coordinate of the vertex is given by  -B/(2A) . In our case the  p  coordinate is   3.0000  

 Plugging into the parabola formula   3.0000  for  p  we can calculate the  y -coordinate : 
  y = 1.0 * 3.00 * 3.00 - 6.0 * 3.00 - 7.0
or   y = -16.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = p²-6p-7
Axis of Symmetry (dashed)  {p}={ 3.00} 
Vertex at  {p,y} = { 3.00,-16.00} 
 p -Intercepts (Roots) :
Root 1 at  {p,y} = {-1.00, 0.00} 
Root 2 at  {p,y} = { 7.00, 0.00} 

Solve Quadratic Equation by Completing The Square

 7.2     Solving   p²-6p-7 = 0 by Completing The Square .

 Add  7  to both side of the equation :
   p²-6p = 7

Now the clever bit: Take the coefficient of  p , which is  6 , divide by two, giving  3 , and finally square it giving  9 

Add  9  to both sides of the equation :
  On the right hand side we have :
   7  +  9    or,  (7/1)+(9/1) 
  The common denominator of the two fractions is  1   Adding  (7/1)+(9/1)  gives  16/1 
  So adding to both sides we finally get :
   p²-6p+9 = 16

Adding  9  has completed the left hand side into a perfect square :
   p²-6p+9  =
   (p-3) • (p-3)  =
  (p-3)²
Things which are equal to the same thing are also equal to one another. Since
   p²-6p+9 = 16 and
   p²-6p+9 = (p-3)²
then, according to the law of transitivity,
   (p-3)² = 16

We'll refer to this Equation as  Eq. #7.2.1  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (p-3)²   is
   (p-3)^2/2 =
  (p-3)¹ =
   p-3

Now, applying the Square Root Principle to  Eq. #7.2.1  we get:
   p-3 = √ 16

Add  3  to both sides to obtain:
   p = 3 + √ 16

Since a square root has two values, one positive and the other negative
   p² - 6p - 7 = 0
   has two solutions:
  p = 3 + √ 16
   or
  p = 3 - √ 16

Solve Quadratic Equation using the Quadratic Formula

 7.3     Solving    p²-6p-7 = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  p  , the solution for   Ap²+Bp+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B²-4AC
  p =   ————————
                      2A

  In our case,  A   =     1
                      B   =    -6
                      C   =   -7

Accordingly,  B²  -  4AC   =
                     36 - (-28) =
                     64

Applying the quadratic formula :

               6 ± √ 64
   p  =    —————
                    2

Can  √ 64 be simplified ?

Yes!   The prime factorization of  64   is
   2•2•2•2•2•2 
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).

√ 64   =  √ 2•2•2•2•2•2   =2•2•2•√ 1   =
                ±  8 • √ 1   =
                ±  8

So now we are looking at:
           p  =  ( 6 ± 8) / 2

Two real solutions:

p =(6+√64)/2=3+4= 7.000

or:

p =(6-√64)/2=3-4= -1.000

Two solutions were found :

1.  p =(15-√253)/2=-0.453
2.  p =(15+√253)/2=15.453