PC - Functions Name ________________________

Completing the Square - Notes Vocabulary: Completing the square is the process of adding a constant """ to the expression #$ + &#to make it a perfect square trinomial. Examples of perfect square trinomials: #$ + 8# + 16 = (# + 4)$ #$ + 6# + 9 = (# + 3)$ #$ + 4# + 4 = (# + 2)$ The quadratics above already have a “c” value that makes it a perfect square trinomial. Look for a pattern between the “b” and “c” value of each. What do you notice? When the “a” of a quadratic is 1, we can find a perfect “c” value by (2$)

$. If you notice, half of the “b” value squared always gives us the perfect “c.” Try it on your own for these examples. Find the perfect “c” and factor. #$ + 12# + ____ = #$ + 16# + ____ = #$ − 18# + ____ = Did this work with a negative “b?” So, now you know how to come up with the perfect “c” value. We can use completing the square to either solve a quadratic that’s not factorable (or is for that matter) or to find vertex form. The rule is, whatever you add to one side of an equation, you must __________________________________.

Finding Vertex Form: 5(#) = #$ + 10# + 13 5(#) − 13 = #$ + 10# + ________ 5(#) − 13 + 25 = #$ + 10# + 25 5(#) + 12 = #$ + 10# + 25 5(#) + 12 = (# + 5)$ 5(#) = (# + 5)$ − 12 Solving using completing the square: 52 = #$ + 14# − 26 78 = #$ + 14# + ________ 78 + 49 = #$ + 14# + 49 127 = #$ + 14# + 49 127 = (# + 7)$ ±√127 = ;(# + 7)$ ±√127 = # + 7 # = −7 ± √127 Modify the above work if “a” is not 1. What do you think you have to do? Try it on the following examples. If you just see f(x), this means write a new function. Otherwise, solve. 5(#) = 2#$ + 24# − 36 77 = 3#$ − 18# + 33

1. Notice that we have a “c,” but it’s not perfect. First step is to always move the “c” to the other side.

2. Find the perfect “c” for the remaining terms. Add this to both sides to balance the equation.

3. Factor the trinomial into a perfect square binomial.

4. Finish writing vertex form by getting f(x) by itself.

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+26 +26

1. Notice that we have a “c,” but it’s not perfect. First step is to always move the “c” to the other side.

2. Find the perfect “c” for the remaining terms. Add this to both sides to balance the equation.

3. Factor the trinomial into a perfect square binomial.

4. This time we solve for “x” by undoing all the operations surrounding “x.”

5. Reduce the radical if possible.

Example √8 = 2√2

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