SOLUTION: solve the equation by completeing the square a2(squared)-12a+27=0

Question 86802: solve the equation by completeing the square
a2(squared)-12a+27=0
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
solve the equation by completing the square
a^2-12a+27=0
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a^2-12a+? = -27+?
Complete the square.
a^2-12a+(12/2)^2 = -27 + (12/2)^2
(a-6)^2 = -27 + 36
(a-6)^2 = 9
(a-6) = 3 or (a-6) = -3
a = 9 or a = 3
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Cheers,
Stan H.
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
note: I'm going to use "x" instead of "a" as my variable
We can convert any quadratic to standard vertex form by this procedure:
Start with the given quadratic

Subtract from both sides

Factor out the leading coefficient
Now to complete the square on the right side we must take half of the x coefficient (in its b) and square it (i.e. )

Take half of and square it (ie ). Add the result() just inside the parenthesis.

This completes the square on the right side. So it goes from and factors to which is a perfect square

Factor the right side into a perfect square
Since we added inside the parenthesis, we really added to the entire right side (just distribute the leading coefficient and you'll see it). So we must add to the other side to balance the equation.

Add to the other side

Multiply
Reduce

Combine like terms on the left side

Reduce any fractions left side
Subtract from both sides
So the quadratic is completed to which is now in vertex form (which is ) where (the stretch/compression factor), (the x-coordinate of the vertex), and is the y coordinate of the vertex. So this means the vertex is (,). Also, since the axis of symmetry is the vertical line through the vertex, the axis of symmetry is (it is equal to the x-coordinate of the vertex).
Here are the graphs of original quadratic and our answer in vertex form
graph of with the vertex (,) and the axis of symmetry (it is the vertical line through the vertex)

graph of with the vertex (,) and the axis of symmetry (it is the vertical line through the vertex)

Notice the two graphs are equivalent; this verifies our answer.

Now to solve for x, we simply need to isolate x:
Set y equal to zero to solve for x
Add to both sides
Take the square root of both sides
Take the square root
Add to both sides

So it breaks down to this
or

So our solution is

or

Notice if you look back at the graph, you will see the roots and . This verifies our answer
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