question form

Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.

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To complete the square on an expression of the form

x^2 + b x

you add and subtract the square of b / 2.

For example to complete the square on

x^2 - 12 x

you can easily see that b = -12, so b/2 = -12/2 = -6.

Squaring this you get 36.

So starting with the original x^2 - 12 x you will add and subtract 36, obtaining

x^2 - 12 x + 36 - 36.

This is completely equivalent to x^2 - 12 x, since 36 - 36 = 0.

The advantage to the form x^2 - 12 x + 36 - 36 is that it now contains the expression (x^2 - 12 x + 36), which we can easilyi factor to obtain the perfect square (x-6)^2.

So we write x^2 - 12 x + 36 - 36 as

(x^2 - 12 x + 36) - 36

and factor the expression in parentheses, obtaining

(x-6)^2 - 36.

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If our original quadratic function was, say

y = x^2 - 12 x + 20

then we could group the x^2 - 12 x, putting the equation into the form

y = (x^2 - 12 x) + 20.

Since completing the square on x^2 - 12 x gives us (x-6)^2 - 36, as we've just seen, our equation becomes

y = ( (x-6)^2 - 36 ) + 20,

which we simplify to the form

y = (x-6)^2 - 16.

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This function could then be graphed by starting with the basic function y = x^2, horizontally shifting the graph 6 units then vertically shifting it -16 units, obtaining a parabola with the same shape as that of y = x^2 but with vertex as (6, -16).

*@

question form

Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.

** Question Form_labelMessages **

** **

** **

@&

To complete the square on an expression of the form

x^2 + b x

you add and subtract the square of b / 2.

For example to complete the square on

x^2 - 12 x

you can easily see that b = -12, so b/2 = -12/2 = -6.

Squaring this you get 36.

So starting with the original x^2 - 12 x you will add and subtract 36, obtaining

x^2 - 12 x + 36 - 36.

This is completely equivalent to x^2 - 12 x, since 36 - 36 = 0.

The advantage to the form x^2 - 12 x + 36 - 36 is that it now contains the expression (x^2 - 12 x + 36), which we can easilyi factor to obtain the perfect square (x-6)^2.

So we write x^2 - 12 x + 36 - 36 as

(x^2 - 12 x + 36) - 36

and factor the expression in parentheses, obtaining

(x-6)^2 - 36.

*@

@&

If our original quadratic function was, say

y = x^2 - 12 x + 20

then we could group the x^2 - 12 x, putting the equation into the form

y = (x^2 - 12 x) + 20.

Since completing the square on x^2 - 12 x gives us (x-6)^2 - 36, as we've just seen, our equation becomes

y = ( (x-6)^2 - 36 ) + 20,

which we simplify to the form

y = (x-6)^2 - 16.

*@

@&

This function could then be graphed by starting with the basic function y = x^2, horizontally shifting the graph 6 units then vertically shifting it -16 units, obtaining a parabola with the same shape as that of y = x^2 but with vertex as (6, -16).

*@