 One way to extend the usefulness of our ability to solve power equations is through the use of something called completing the square. What about equations like x squared plus 6x equals 16? Now if you're working this problem and suddenly heard a voice say add 9 to both sides. What you did at that point would depend on how often you listen to these little voices going off in your head. Eat more chocolate. But in this case suppose we actually listen. If we add 9 to both sides we have and what's rather remarkable is the left-hand side is a perfect square x plus 3 quantity squared. And so this allows us to solve the equation because now we have a power equation x plus 3 is plus or minus the square root of 25 and we can solve it. And so what this tells us is the 9 made our expression a perfect square. We say that it completed the square. So what we need is a little voice telling us what we need to add to complete the square. Of course the problem with listening to little voices inside our head is that it can get a little bit noisy. So let's see if we can do something more mathematical. So we might start with the observation that if we have the square of a binomial it's going to look like. And if we look at this carefully we see the following. A trinomial will be a perfect square of a binomial if the x squared and constant terms are both perfect squares. The x term is twice the product of the square roots of the x squared and constant terms. So for example, we might try to figure out whether 4x squared plus 8x plus 16 is a perfect square. So first we'll check to see if the x squared and constant terms are perfect squares. So our x squared term 4x squared is 2x quantity squared. So it's a perfect square. Our constant term is 16, which is 4 squared. So it's also a perfect square. And so now our x term should be twice the product of the square roots of the x squared and the constant terms. So let's check it out. The square roots of the x squared and the constant terms are going to be 2x and 4. So we check 8x is it 2 times 2x times 4? And the answer is no. So 4x squared plus 8x plus 16 is not a perfect square. So can we make something into a perfect square? So let's find c so that 9x squared plus 12x plus c squared is a perfect square and then we'll complete the square. So remember that the trinomial is going to be a perfect square only if the x squared and the constant terms are perfect squares. And the x term is twice the product of the square roots of the x squared and the constant terms. So our x squared term 9x squared is a perfect square because it's the square of and our constant term, the term that doesn't include the variable at all, is a perfect square because it's the square of c. Now we want to complete the square, which means that our x term has to be twice the product of the square roots of the x squared and constant term. So 12x has to be equal to 2 times 3x times c. And now we have an equation that we can solve for c. So 12x should be 2 times 3x times c. Associativity and commutativity of multiplications as we can rearrange these factors any way we want to. So we'll rearrange them this way. And now let's look at our equation. It says 12 times x is the same thing as 6c times x. And so that means we want 12 to be equal to 6c. And so c itself has to be equal to 2. So c squared is 2 squared, otherwise known as 4. And so 9x squared plus 12x plus 4 is going to be the square of 3x plus 2. The important version of this problem is known as completing the square. So here we have the first two terms in something that's supposed to be a perfect square. And so the question is what constant can we add to make this into a perfect square? So we want to find c squared so that x squared plus x plus c squared is a perfect square. So we might begin by noting that x squared is a perfect square x squared. c squared is a perfect square c squared. And we want our x term to be 2 times the square roots of our x squared and constant terms. So it should be 2 times x times c. And so we can solve for c. We'll rearrange terms a little bit. And so we want 1 times x to be the same as 2c times x. And so that means 1 is equal to 2c. And so c itself is equal to 1 half. And so that says if we add 1 half squared to x squared plus x, we're going to get a perfect square, namely the square of x plus 1 half.