Completing the square
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.