what is the quadratic function

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(k+1)(k-5)=0

May 13th, 2015

k+1)(k-5)=0 

=>k+1=0;or k-5=0 

=>k=-1;k=5.

hence

Solve Quadratic Equation by Completing The Square

 4.2  Solving  k2-4k-5 = 0 by Completing The Square .
Add   to both side of the equation : 
  k2-4k = 5

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

Add  4  to both sides of the equation :
 On the right hand side we have :
  5 +  4 or,  (5/1)+(4/1) 
 The common denominator of the two fractions is  1  Adding  (5/1)+(4/1)  gives  9/1 
 So adding to both sides we finally get :
  k2-4k+4 = 9

Adding  4  has completed the left hand side into a perfect square :
  k2-4k+4  =
  (k-2) • (k-2) =
 (k-2)2 
Things which are equal to the same thing are also equal to one another. Since
  k2-4k+4 = 9 and
  k2-4k+4 = (k-2)2 
then, according to the law of transitivity,
  (k-2)2 = 9

We'll refer to this Equation as  Eq. #4.2.1  

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

Note that the square root of
  (k-2)2   is
  (k-2)2/2 =
 (k-2)1 =
 k-2


Now, applying the Square Root Principle to  Eq. #4.2.1  we get:
  k-2 = √ 9 

Add   to both sides to obtain:
  k = 2 + √ 9 

Since a square root has two values, one positive and the other negative
  k2 - 4k - 5 = 0
 has two solutions:
 k = 2 + √ 9 
 or
 k = 2 - √ 9 

Solve Quadratic Equation using the Quadratic Formula

 4.3  Solving  k2-4k-5 = 0 by the Quadratic Formula .
According to the Quadratic Formula, k , the solution for  Ak2+Bk+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
 
 - B ±  √ B2-4AC
 k =  ————————
 2A  In our case, A =  1
 B =  -4
 C =  -5 Accordingly, B2 - 4AC =
 16 - (-20) =
 36Applying the quadratic formula :

 4 ± √ 36 
 k = —————
 2Can  √ 36 be simplified ?

Yes! The prime factorization of  36  is
  2•2•3•3 
To be able to remove something from under the radical, there have to be   instances of it (because we are taking a square i.e. second root).

 36  = √ 2•2•3•3  =2•3•√ 1  =
 ± 6 • √ 1  =
 ± 6 


So now we are looking at:
  k = ( 4 ± 6) / 2

Two real solutions:

k =(4+√36)/2=2+3= 5.000 

or:

k =(4-√36)/2=2-3= -1.000 

Two solutions were found :

  1.  k = 5
  2.  k = -1

May 13th, 2015

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