 So now let's see if we can solve equations by completing the square. So let's try to solve x squared plus 8x equals 15. So we want to find c where x squared plus 8x plus c squared is a perfect square. And we note x squared is a perfect square. c squared is also a perfect square. And so we want to make sure that 8x is 2 times x times c. And solving for c, again, we can rearrange these products any way we want to. So 8x is 2c times x. So 8 times x is the same as 2c times x. And so we want 8 to be equal to 2c. So c is equal to 4, and c squared is 4 squared, otherwise known as 16. So that says if I add 4 squared to x squared plus 8x, we'll get a perfect square. Well, in math, you can get anything you want as long as you pay for it. So here's my equation x squared plus 8x equals 15. I want a plus 4 squared so I can buy it as long as I pay for it. In this case, because it's an equation, I can do the same thing to both sides of the equation. So I'll add 4 squared. On the left-hand side, I get x squared plus 8x plus 4 squared. Over on the right-hand side, I get 31. Now remember the reason we did this was to make sure that our left-hand side was a perfect square. And it is our left-hand side is the square of x plus 4. And now we have a power equation. Square equals number. And so that tells us that x plus 4 is a square root of 31. Either x plus 4 is the principal square root of 31, or x plus 4 is minus the principal square root of 31. Solving each equation separately, which gives us two solutions to this equation. How about 4x squared equals 25 plus 12x? The first thing to recognize here is that if we have any hope that this is going to be square equals number, we have to have equals number. And so this equals 25 plus 12x won't work. So we'll rearrange things. We'll subtract 12x from both sides and get something equals number. Now we want to find c, where 4x squared minus 12x plus c squared is a perfect square. So we know that 4x squared is the same as 2x quantity squared. c squared is, of course, c squared. And we want to make sure that minus 12x is 2 times the square roots, 2x times c. And so now we can solve for c. And we find that c is equal to negative 3. So c squared is negative 3 squared. So we'll add this to both sides of our equation. Over on the right-hand side we get 34. Over on the left-hand side, if we've done everything correctly, what we should get is the square of 2x plus c. So the left-hand side should be the square of 2x minus 3. And so now my equation is in the form square equals number. So I know 2x minus 3 is either the principal square root of 34 or negative the principal square root of 34. And we can solve these two equations separately, which gives us two solutions to this equation. How about something like 3x squared plus 5x equals 8? Well, there's actually three different ways we can solve this by completing the square. Pick which one you like. Now, if we don't want to think about this problem at all, we can proceed as follows. We want to find c, where 3x squared plus 5x plus c squared is a perfect square. And so 3x squared needs to be the square of something. So it's got to be the square of the square root of 3x. Now, there's a little bit of a problem here. Because we've introduced this square root of 3 here, it's going to show up in all the rest of our problem. And if we proceed in the standard way from this point, we end up with something like this. We figure out what c is. We add c squared to both sides, which gives us a perfect square. We take the square root of both sides, remembering to use our plus or minus principle square root, and then we solve the whole mess for x. So if you don't mind ending up with solutions that look like this, you can solve this equation without giving it any additional thought. On the downside, an expression like this is hard to work with. So maybe we can try to end up with a simpler expression if we give our equation a little bit of additional thought beforehand. And having solved this equation this way, the thing that we might notice is that the problem started to arise at this first step, where we needed 3x squared to be a perfect square. And the reason that became a problem is that forced us to introduce the square root of 3. So let's see if we can avoid that. So one thing we could do is if we divide every term by 3, that will eliminate the coefficient of x squared. Now, we still want to complete the square, but this time we're looking for a c where x squared plus 5 thirds x plus c squared is a perfect square. So once again, we'll note that x squared is x squared. 5 thirds x we want to be 2 times x times c, and so 5 thirds x is 2cx, and we can solve this c equals 5,6, adding 5,6 squared to both sides. Again, the left-hand side will be the square of the sum of the roots x and 5,6. The right-hand side will be some arithmetic expression, which we can leave in this form. We have square equals, so we'll take the square root of both sides. So x plus 5,6 is plus or minus the principle square root of 8 thirds plus 5,6 squared. We can solve for x by subtracting 5,6, and we'll get our solution. So again, the basic steps here are the same as we used for completing the square. The only difference is that we did this preliminary division so that our coefficient of x squared would be a1. And that's because we wanted to make sure that our x squared term was an easy perfect square. The only disadvantage here is that we had to work with fractions. And maybe you like working with fractions, which is OK, but maybe you don't, which is also OK, as long as you figure out how to avoid them. So let's see if we can avoid these fractions. So remember, we'd like our x squared term to be a perfect square, and so one possibility is if our coefficient is a perfect square, then we're in good shape. So our coefficient is 3. If we multiply everything by 3, we'll have a perfect square coefficient. And again, we want to find c where 9x squared plus 15x plus c squared is a perfect square. And so we know that our first term, 9x squared is 3x squared. We want 15x to be 2 times 3x times c. So we can solve for c, which will be 15 over 6. So we can add 15 over 6 squared to both sides. Our left-hand side is the square of the sum of the roots 3x plus 15 over 6. And our right-hand side, we'll just leave it on reduced form. We'll take the square root of both sides, and we'll solve for x. Now you might notice we still have to deal with fractions, and we'll leave it as an interesting challenge. Is there something else we can do that completely avoids fractions and think to recognize is why we got that fraction in the first place? It's worth emphasizing that all three of these solutions rely on completing the square. If we decide we're not going to think about the problem at all, we end up with a solution like this. If we decide we're going to think about the problem a little bit, we'll end up with a solution like this. And if we think about the problem a little bit more, we'll end up with a solution like this. So take your pick and choose the solution that you want to work with.