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CCSS Math: HSA.SSE.B.3, HSA.SSE.B.3b

Use completing the square to
find the value of c that makes x squared minus 44x plus c--
so we can just figure out a c-- that makes it a perfect
square trinomial-- and a trinomial is just a polynomial
with three terms here. Then write the expression as
the square of a binomial. So we have x squared
minus 44x plus c. So how do we make this into
a perfect square? Well, if you just look at the
traditional pattern for a perfect square, let's
just think of it in terms of x plus a squared. That's the same thing as x plus
a times x plus a, and we've seen this before. And if you were to multiply this
out, that's x times x, which is x squared, plus
x times a, which is ax. Plus a times x, which is ax. Plus a times a, which
is a squared. So it's x squared plus 2ax,
these two, you have an ax plus an ax gives you 2ax,
plus a squared. So if we can get this into this
pattern, where I have whatever value is here, if
I take half of it, right? This is going to be 2a here. If I take half of it and square
it over here, then this will be a perfect square. So if we look over here, this
thing right here is 2a, if we want to pattern match, if we
want to make this look like a perfect square. That has to be 2a. So negative 44 is equal to 2a. And this, right here, this c, if
we pattern match, c has to be equal to a squared. So what's a? Well, if we know negative 44
is 2a, we can divide both sides of that by 2. And we know that negative 22
has got to be equal to a. a has got to be equal
to negative 22. a is half of the coefficient
right here. It's half of negative 44. And whenever you complete the
square, it's always going to be half of the coefficient
right here. Now, if that's a, what
does c need to be? Well, c needs to be a squared
in order for this to be a perfect square. So c needs to equal negative
22 squared. And we can figure out
what that is. 22 times 22, we could put the
negative later-- actually it's just going to be the same thing
because the negative times a negative
is a positive. 2 times 22 is 44, put a 0. 2 times 22 is 44. Get a 4, get an 8, get a 4. So it's 484. So if we were to rewrite this
as x squared minus 44x plus 484, then this is a perfect
square trinomial. Or we could write
it like this. This is x squared minus 2
times-- or maybe I should write it this way-- plus 2
times negative 22x plus negative 22 squared. And when you view it that way,
it's pretty clear that this is a perfect square, and if you
were to factor it, it's the same thing as x minus
22 times x minus 22, or x minus 22 squared. These are all equivalent
statements.