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Use completing the square to find the value of c that makes

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x squared minus 44x plus c-- so we can just figure out a

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c-- that makes it a perfect square trinomial-- and a

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trinomial is just a polynomial with three terms here.

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Then write the expression as the square of a binomial.

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So we have x squared minus 44x plus c.

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So how do we make this into a perfect square?

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Well, if you just look at the traditional pattern for a

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perfect square, let's just think of it in

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terms of x plus a squared.

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That's the same thing as x plus a times x plus a, and

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we've seen this before.

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And if you were to multiply this out, that's x times x,

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which is x squared, plus x times a, which is ax.

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Plus a times x, which is ax.

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Plus a times a, which is a squared.

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So it's x squared plus 2ax, these two, you have an ax plus

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an ax gives you 2ax, plus a squared.

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So if we can get this into this pattern, where I have

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whatever value is here, if I take half of it, right?

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This is going to be 2a here.

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If I take half of it and square it over here, then this

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will be a perfect square.

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So if we look over here, this thing right here is 2a, if we

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want to pattern match, if we want to make this look like a

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perfect square.

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That has to be 2a.

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So negative 44 is equal to 2a.

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And this, right here, this c, if we pattern match, c has to

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be equal to a squared.

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So what's a?

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Well, if we know negative 44 is 2a, we can divide both

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sides of that by 2.

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And we know that negative 22 has got to be equal to a. a

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has got to be equal to negative 22.

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a is half of the coefficient right here.

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It's half of negative 44.

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And whenever you complete the square, it's always going to

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be half of the coefficient right here.

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Now, if that's a, what does c need to be?

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Well, c needs to be a squared in order for this to be a

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perfect square.

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So c needs to equal negative 22 squared.

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And we can figure out what that is.

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22 times 22, we could put the negative later-- actually it's

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just going to be the same thing because the negative

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times a negative is a positive.

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2 times 22 is 44, put a 0.

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2 times 22 is 44.

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Get a 4, get an 8, get a 4.

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So it's 484.

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So if we were to rewrite this as x squared minus 44x plus

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484, then this is a perfect square trinomial.

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Or we could write it like this.

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This is x squared minus 2 times-- or maybe I should

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write it this way-- plus 2 times negative 22x plus

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negative 22 squared.

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And when you view it that way, it's pretty clear that this is

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a perfect square, and if you were to factor it, it's the

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same thing as x minus 22 times x minus 22,

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or x minus 22 squared.

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These are all equivalent statements.

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