Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. I'm going to solve by taking the square root of both sides. So the first thing we want to do is we want to get the variable on one side by itself. So we're going to add 16 to each side and get X ^2 equaling 16. When we get rid of that square, we're going to remember positive and negative. So we're going to have positive and negative sqrt 16 because the way we get rid of a square is we square root. And we all know that sqrt 16 is really just four. So positive and -4 for the second one, we're going to take that 16 to the other side. And we're going to get -16 when I get rid of the square, I square root it. And if I square root one side, we're going to square root the other, remembering plus and minus. So sqrt X ^2 is X sqrt a negative is I, and sqrt 16 is 4. This next one, when we square root it, we're going to have sqrt X - 3 ^2 equaling positive or negative sqrt 25 S The square root and the square are going to cancel X - 3 equal positive, and -5. So this is really two different equations. This is really equation X - 3 equal 5 and X - 3 equal -5 SX is going to equal 8 or X is going to equal -2. This next one, we're going to square root each side, so sqrt X + 5 ^2 is going to equal the positive negative sqrt -16. So the square root and the square cancel. The square root of a negative is I and sqrt 16 is 4, so we have X + 5 equaling 4I and X + 5 equal -4 I. So then we're going to just subtract the five so we get -5 + 4 I and -5 -, 4. I remember we put them in a plus BI form. When we're talking about complex numbers. The last one on this page, we actually are going to want to start by factoring X ^2 -, 14 X plus 49 is X -, 7 quantity squared. Now how do we get rid of that square? We're going to square root it. If I square root one side, we're going to remember to square root the other, also remembering plus and minus. So the square root in the square cancel. So we get X -, 7 equaling positive or negative square roots of seven. So X - 7 = sqrt 7, X -7 equal negative sqrt 7. When we add the seven to the other side, our final answers are going to be X = 7 plus root 7 and X = 7 minus root 7. The next time, the next type of problem. So how do we complete a square? We're going to take the middle term, the middle coefficient, and take half of it, middle coefficient divided by two, and we're going to square it. So middle coefficient 6 / 2 ^2, that's going to give me X ^2 + 6 X +9. And we know that that factors into X + 3 quantity squared. So we're going to take half of the 10 and square it. So y ^2 + 10 Y half of 10 is five, 5 ^2 is 25. So that factors into y + 5 quantity squared. This one, we're going to take half of -5 and square it. So X ^2 -, 5 X plus 25 fourths, that's going to factor into X -, 5 half squared. This last one, we're going to take half of negative 1/3. One way of thinking about taking half is I could actually take that number and put A2 in the bottom because half of -1 third and we're going to square it. So that's going to be Z ^2 -, 1/3 Z negative 1/3 * 1/2 is -1 sixth, and then I want to square that. So -1 ^2 is one, 6 ^2 is 36. That's going to factor into Z -, 1 six quantity squared. Now the interesting thing that occurs here is once I simplify, the 6 / 2 is really that three when it's in factored form. The 10 / 2 is really that five in factored form. The -5 halves -5 halves, negative 1/6, negative 1/6. So we need that half the middle term squared to know what to add. But when we factor it, it's really just half the middle terms and then we do the quantity squared. So on these next ones, we're going to think about half of -10 ^2. Well, half a -10 ^2 is going to be +25. So when we factor it, it'll be t -, 5 half of 3 ^2, so it's going to be 9 fourths. And when we factor it, we're going to have X + 3 halves, half of the -5 sixths, which I can really think of multiplying by two in the denominator, it's going to be 25 twelfths. And over here, when we factor, it's going to be -5 twelfths. We already had the minus there, so we're going to continue by completing the square. Now we're actually going to solve problems. So if I have X ^2 + 6, XI now want half of that six and quantity squared. And if I do it to one side, we have to do it to the other. So the left hand side turns into X ^2 + 6 X +9, and the right turns into 7 + 9 X ^2 + 6. X plus nine turns into X + 3 quantity squared when we factor it, and 7 + 9 is 16. How do we get rid of a square? We square root it, and if I square root one side, we're going to square root the other. Remembering positive and negative sqrt 16 is just four, so we're going to get X + 3 equal four, X + 3 equal -4. Subtract the three from each side and we get X equal 1 and X equal -7. This next one, we're going to start by taking the 21 to the other side to start. Then we're going to take half of that -10 and square it. So we're going to add -10 / 2 ^2. If I do it to one side, we have to do it to the other. So this is going to turn into t ^2 -, 10, T plus 25 equaling -21 + 25. The left side's going to factor into t -, 5 quantity squared, and the right side's going to combine like terms to give me 4 when I square root it. We have to remember to square root both sides and sqrt 4 is 2. So we're going to get t - 5 equaling 2 and t - 5 equaling -2. If I add five, I get T equal 3 and T equal. Oops, T equals seven 5 + 2 and 5 + -2 is 3. We're going to continue with completing the square. So X ^2 -, X, add the six to the other side, take half of the middle coefficient, which in this case is an understood -1 and square it. If I do it to one side, I need to do it to the other. So this left side's going to turn into X -, 1/2, quantity squared. The right side, we're going to get 6 + 1/4. Well, 6 is really 24 fourths plus 1/4. So the right side's going to turn into 25 fourths. So when we square one square root one side, we're going to square root the other. We get X -, 1/2 equaling positive and negative. Sqrt 25 is 5, sqrt 4 is 2. So X - 1/2 equal 5 halves and X -, 1/2 equal -5 halves. If I add a half across, 1/2 + 5 halves is 6 halves or three. If I add a half across -5 halves plus 1/2 is -4 halves or -2 this next one I need to get all the A's on one side, so a ^2 -, 4 A equal -2. Then I'm going to take half of the -4 and square it. If I take half of the -4 and squared on one side, I need to take half of the -4 and squared on the other. So we get a ^2 -, 4 A+ 4 equaling -2 + 4 A -2 quantity squared equal 2. How do we get rid of a square? We square root it. And if I square root one side, we've got to remember positive, negative and square root the other. So we get a - 2 = sqrt 2, A -2 equal negative sqrt 2. So a is 2 plus root 2 and a = 2 minus root 2. And this next one, we have a coefficient in the front. So the very first thing we have to do is we're going to take the entire equation and divide by whatever that leading coefficient is. So I'm going to take four X squared and divide it by 4. I'm going to take 8X and divide it by 4. I'm going to take -3 and divide it by 4. So I now have a new equation that says X ^2 + 2 X equal negative 3/4. Now I'm going to take half of that leading coefficient 2 / 2, and I'm going to square it. And if we do it to one side, we have to do it to the other. So we get X ^2 + 2 X plus one equaling negative 3/4 + 1. So X + 1 quantity squared. We just factor it negative 3/4 + 1. One is really 4 fours, so negative 3/4 + 1/4 is going to give us sorry, negative 3/4 + 4 fourths. Think negative 3/4 + 4 fourths. So -3 + 4 is 1/4. Now we're going to square root each side, so X + 1 is going to equal positive negative square root 1/4. So X + 1 = sqrt 1 is 1. Sqrt 4 is 2X plus one also equals -1 half. If I subtract the one, 1/2 - 1 would give me a negative 1/2. If I subtract the one for this other possibility, we get X equal -3 halves. Let's do another couple of these. So we're going to divide everything through by two. We're going to get X ^2 -, 5 halves X, and we're going to take that three to the other side. So three halves. We're going to take half of the middle coefficient. When I say half of, I can also think of it as multiplying the denominator by two and squaring it. If we do it to one side, we have to do it to the other side. So we have X ^2 -, 5 halves, X + 25 sixteenths equal. If I wanted to get a common denominator between three halves and 6:25 sixteenths, it would be 16. So we're going to have X -, 5 fourths quantity squared, equaling 16, so 24 sixteenths. If I think about multiplying the top and the bottom each by 8 to get a common denominator of 16 plus the 25 sixteenths, so X -, 5 fourths squared equal 49 sixteenths. We're going to square root. So we get X -, 5 fourths equaling positive. Sqrt 49 is 7 and sqrt 16 is four X -, 4 is awesome, X -, 5 fourths is also going to equal -7 fourths. If I add five fourths, 7 fourths plus 5 fourths is 12 fourths, or three -7 fourths plus 5 fourths is -2 fourths or negative 1/2. So our 2 answers there are three and negative 1/2 for the next one. We're going to start by dividing everything through by 4. So we're going to get X ^2 + 3/4 X equaling 5 fourths. I'm going to take half of the middle coefficient and square it. So 3 / 4 * 2 ^2. That's a three. OK, That's going to equal 5 fourths plus 3 / 4 * 2 ^2. So X ^2 + 3 fourths, X + 960 fourths. If I want to get a common denominator between 5 fourths and 960 fourths, I'm going to have to multiply that four by 16, because 16 * 4 is 64. And if I do the bottom by 16, we have to do the top. So that's going to turn into 8060 fourths plus 960 fourths. So X + 3/8 ^2 is going to equal 8960 fourths. So we're going to square root. We're going to get X + 38 three eighths equals positive and negative sqrt 89 sqrt 64 is 8. So we have X + 3/8 equaling sqrt 89 eighths and X + 3/8 equaling negative sqrt 89 eighths. So X equal negative 3/8 plus sqrt 89 eighths and X equal negative 3/8 -, sqrt 89 eighths. So a bonus for tonight. Prove the quadratic formula by completing the square on the statement. So do what we did before. Your first step is going to be to divide each term by a. Thank you and have a wonderful day.