SOLUTION: Bill used completing the square to find the zeroes of the function {{{y=9x^2-12x-33}}}. How is this done ?

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Question 120255: Bill used completing the square to find the zeroes of the function . How is this done ?
Found 2 solutions by MathLover1, Edwin McCravy:
Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


Start with the given equation



Add to both sides



Factor out the leading coefficient



Take half of the x coefficient to get (ie ).


Now square to get (ie )





Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of does not change the equation




Now factor to get



Distribute



Multiply



Now add to both sides to isolate y



Combine like terms




Now the quadratic is in vertex form where , , and . Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation we get:


Graph of . Notice how the vertex is (,).



Notice if we graph the final equation we get:


Graph of . Notice how the vertex is also (,).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!
SOLUTION BY EDWIN:
Bill used completing the square to find the zeroes of the function
. How is this done ?

It's long! Sorry!

y = 9x² - 12x - 33

To find the zeros we substitute 0 for y and solve for x:

0 = 9x² - 12x - 33

Let's put the 0 on the right:

9x² - 12x - 33 = 0

1. Isolate the terms in x on the left:

That is, add 33 to both sides:

    9x² - 12x = 33

2. Divide every term through by the coefficient
   of x², since it is not 1.

    x² - x = 

Simplifying:

    x² - x = 

3. To the side multiply the coefficient of x by 

      =  = 

4.  Square the result of step 3:

      = 

5.  Add that to both sides of the equation we 
    had at the end of step 2:

    x² - x +  = +

6.  Factor the left side:

    {x - )(x - ) = +

7.  Combine the terms on the right sides

    Write  as 

    {x - )(x - ) = +

    {x - )(x - ) = 

8. Write the left side as a square.  We always can.  That's
   why the method is called "completing the square".

    {x - )² = 
  
10. Use the principle of square roots:

       x -  = ±

11. Solve for x.

                   x =  ± 

That can be written:

                   x = 

Edwin
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