complete the square and find the vertex for of the quadratic function

Question

F (x)=x²-8x+69

Answer 1

Hi, again..Jodi

 

f(x)=X^2-8x+69

 

There is no real solution for this problem either.  Let me tell you how I know.  Look at the quadratic equation.  a=1, b=8, c=69.  Now, there is a concept called the DETERMINANT, the determinant tells you if you have two real solution, one real solution, or no real solution.  These are the rulues

 

b^2-4ac = 0     You get one real solution

b^2-4ac> 0      You get two real solutios

b^2-4ac< 0      You get no real solution

 

In this case, a=1, b=-8, c=69.  Plug this numbers into the formula for the determinant:

b^2-4ac

(-8)^2 -4(1) (69)

64-276< 0

-212 < 0          You have no real solution.  But I digress.  Lets go on with the problem.

 

Orininal Problem:

0              =     X^2-8x+69

-69                         -69     

-69           =     X^2-8x                     Divide the b term, -8 by 2 and square it. (-8/2)= -4. (-4)^2= 16

+16                         +16              Add 16 to both sides

-53           =     x^2-8x+16

-53           =    (x-4)^2                     Factoring

+/- 53i      =     x-4                          Taking the sq. rt. of both sides. 

+/- 53i +4 = x

 

So, one again, you have two answers, since they are not real I used i:

a) 53i+4

b) -53i+4

 

Now, to find the vertex use [-(b)]/[2(a)]       Remember a=1, b=-8

                                        [-(-8)]/[2(1)]   

                                        [8]/[2]

                                           4                   This is the value of the x coordinate of the vertex.

 

Lastly, to find the y coordinate of the vertex, plug in 4 for x into x^2-8x+69

                                                                                            (4)^2-(8)(4)+69

                                                                                             16-32+69

                                                                                                  53

 

So, the coordinate of the vertex is (4, 53)

 

Ok.  We are done.

 

D.Y. Taylor              

Answer 2

f(x)=x²-8x+69          Take half the coefficient of the "x" term and square it.

-8 = -4     (-4)²=16   Add and subtract this number from your function.

 2

 

f(x)=(x²-8x+16)+69-16    Factor the quadratic.

f(x)=(x-4)(x-4)+53

F(x)=(x-4)²+53

 

Vertex: (4,53)