Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Completing the square solved by taking the square root of both sides. So on these problems, we're going to get the variable on one side by itself and everything else on the other. Then we're going to square root and we need to take the positive and the negative when we square root. So X is positive or -4. The fact that it's a two as its exponent tells us we should expect 2 answers this next one. When we take 16 to the other side and square root it, we can see we have positive negative sqrt -16 and that's going to be 4I. This next one, when we square root, we're going to get positive, negative sqrt 25, which is really five, but we need to get the X by itself, so we're going to add 3. So 3 ± 5, three plus five is eight. 3 - 5 is negative two X + 5 ^2. So we square root it. We remember that positive and negative sqrt -16 was 4I. We talked about it right back here. So X is going to equal -5 ± 4 I. We can't combine those because they're not like terms. So -5 + 4 I is 1 answer and -5 -, 4 I is the other answer. For this next one, we're actually going to complete the square 1st and realize these three terms on the left factor into X -, 7 ^2. So then when we square root to get rid of the square, we get X -, 7 equaling positive negative root 7, or X is 7 plus or minus root 7. Now that's two answers. That's seven plus root 7 and 7 minus root 7. There's some for you to try. The next type of problem we're going to do is completing the square. We need to make sure there's a one coefficient on the X ^2 term or the squared term, and then we're going to take half of the middle coefficient or half of the linear coefficient and then square it. So half of 6 is going to be 3 and 3 ^2 is going to be 9. So X ^2 + 6 X +9, and that would factor into the quantity X + 3 ^2. So the whole purpose is to find a number we can put here to make it into a quantity squared. So half of 10 is five and 5 ^2 is 25. So that's going to factor into y + 5 ^2. That half of 10 is always going to end up there if I've done it correctly. So half a -5, we're just going to leave it as a fraction. So five -5 halves squared, 25 fourths. So when I factor it, X -, 5 halves, quantity squared, half of -1 third is going to be negative 1/6. When we square it, we get 136. So Z -, 1 six squared. If I want -1 third and I want half of it, I could think of multiplying it. So that's the negative 1/6. Here's some for you to try. We're going to complete the square by solving. So this time we have X ^2 + 6. X equals seven. We're going to add half of that 6 ^2, so 9. But this one's an equation. So if we add 9 to one side, we're going to add 9 to the other. That's going to turn into X + 3 quantity squared equaling 16. When we square root it, we get X + 3 equaling positive and negative squared to 16, which is 4. So -3 ± 4 X is going to be 1 or -7. For this next one, we're going to take the 21 to the other side to start. So t ^2 -, 10 T equal -21 and take half a -10 and square it. So we're going to add 25 to each side. So we're going to get t -, 5 quantity squared equaling 4. When we square root each side, square to four is 2. So T is 5 ± 2 or seven and three. This next One X ^2 -, X, we're going to take six to the other side. Half of one is 1/2 ^2 is 1/4. If I add a fourth to one side, we're going to add it to the other. So X -, 1/2 quantity squared. Now six we could think of as 24 fourths plus 1/4 would give me 25 fourths. So now we get X -, 1/2 equaling positive and negative. Sqrt 25 is 5, sqrt 4 is 2. So if we have 1/2 + 5 halves, that's six halves or three. And if we have 1/2 -, 5 halves, that's -4 halves or -2. This next one, we need to get all of the A's on one side. So we're going to have a ^2 -, 4 A. We're going to take half a -4 and square it. So plus 4 equal -2 + 4. So we're going to get a - 2 quantity squared equaling 2. So a - 2 equaling positive negative sqrt 2 A equal to plus or minus sqrt 2. Can't combine those, they're not like terms. If the coefficient in front is not equal to one, we have to start by dividing everything through by the coefficient. So X ^2 + 2 X equal -3 fourths. Then half of two is 1 and one squared is one. We're going to add 1 to each side. So the left turns into X + 1 quantity squared, the right turns into a fourth. When we square root, we get positive negative sqrt 1 fourth, which is 1/2 negative 1 ± 1/2. So we get -1 + 1/2 or negative 1/2 and -1 -, 1/2 negative 3 halves. This next one we're going to divide by two, so we get X ^2 - 5 halves, X -, 3 halves. Actually, let's take that -3 halves to the other side. So we're going to take half of five halves, which would be 5 fourths, and then we're going to square it. So we're going to get 25 sixteenths. And if we add it to one side, we're going to add it to the other. So we get X -, 5 fourths, quantity squared equaling. If I multiply by 8 / 8, I get 24 sixteenths plus 25 S 49 sixteenths. So X -, 5 fourths equals positive or negative. I can square each top and bottom separately. 7 fourths, so 5 fourths ±7 fourths, 5 + 7 is 12 / 4 is 3, and 5 - 7 is -2 / 4 negative 1/2. So we're going to start by dividing by four X ^2 + 3/4 X equaling -5 fourths. We're going to add it over, actually make it positive. Half a 3/4 is 3 eighths, 3/8 ^2 is going to be 960 fourths. If I add 960 fourths to one side, we're going to add it to the other. Oh, and I knew I was going to run out of room there and 960 fourths, so here's going to be X + 3/8 ^2 equaling 64. So we're going to multiply by 1616 times. Five is 80, so we're going to get 8960 fourths. So X + 3/8 is going to equal positive negative sqrt 89 eighths, X is -3 ± sqrt 89 eighths. Then the last one we're going to give you a couple try. It's the last one is doing it generically, and we're going to start by dividing everything through by an A, and we're going to take that C to the other side. Then we're going to take half of that B / A term and square it. So half of the B / A term squared is going to be b ^2 / 4 A squared. If I add it to one side, I'm going to add it to the other. Then we're going to factor the left side so we get X + b / 2 A squared. On the right side, I'm going to get a common denominator of 4A squared, so b ^2 -, 4 AC, multiplying the top and the bottom by 4A, going to square root each side. So X + b / 2 A equaling positive negative. We can't do anything with sqrt b ^2 - 4 AC because it was two terms. But sqrt 4 is 2 and sqrt a ^2 is a. So negative b / 2 A+ or minus the square root b ^2 - 4 AC over 2A. And when we simplify that, we get X equal negative B plus or minus the square root b ^2 -, 4 AC all over 2A which is actually the quadratic formula. Thank you and have a wonderful day.