Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. The quadratic equation is built from completing the square. So to complete the square here we have always have to have A1 coefficient. So we're going to divide everything through by A and we're going to take that constant term to the other side. Then we're going to take half of the coefficient on the linear term and square it. So half of b / a is b / 2 A. And then when we square it, we get b ^2 / 4 A squared, and if I add it to one side, I'm going to add it to the other. The left side now is going to complete the square to make it X + b / 2 a quantity squared. If we foiled out X + b / 2 A times X + b / 2 a, we get this on the right hand side. I'm going to get a common denominator, so I'm going to multiply the C / a by 4A on top and 4A on bottom. Then to get rid of a square, I'm going to square root it, and when I square root I'm going to take positive and negative of each side. I can't take sqrt b ^2 -, 4 AC because it's not a monomial. There's 2 terms there, but sqrt 4 A squared I can make into two a. Our goal is to get X by itself, so we're going to take this b / 2 A to the other side. So we're going to get negative b / 2 A+ or minus square root b ^2 -, 4 AC all over 2A. And if we just put those over the same denominator, we get the quadratic formula which says X equal negative B plus or minus square root b ^2 -, 4 AC all over 2A. So our examples, we're literally just going to stick things into that formula. So here our A is 2, our B is -3, and our C is 5. SX equals the opposite of B plus or minus the square root b ^2 -, 4 * a * C all over to A. When we simplify that up, we get 3 ± 9 - 49 - 40 would be -31 / 4, so 3 ± sqrt 31 sqrt A negative is I all over 4. We can't combine those terms because one's real and one's imaginary. So that's our final answer. On this next one, we're going to need to take all of the terms to one side because the quadratic formula only works if it equals 0 in the beginning. So our a is one, our B is -6, and our C is 3. So X equal the opposite of b ± sqrt b ^2 - 4 * a * C all over 2A. So the opposite of -6 is 6 ± sqrt 30. Sqrt -6 ^2 is 36 - 1236 - 12 is going to be 24. Now 24 actually could be 4 * 6, and we know sqrt 4, so this one A2 will come out of everything. If I factor 2 out of the top, I'd get 3 plus or minus squared of six all over 2. So our final answer there is going to be 3 plus or minus square roots of six. This next one we're going to have to distribute first. So we get seven X ^2 + 14 X +5 equaling three X ^2 + 3 X. We've got to get everything to one side. So four X ^2 + 11 X plus 5X equal the opposite of B plus or minus the square root b ^2 -, 4 * a * C all over to a so -11 plus or minus square root 121 -8121 -. 80 is going to give me 121 -, 8041 / 8, and 41 is prime. So that's not going to be able to do anything. So the next one we're going to look at is X ^3 + 8, and this one we actually are going to expect 3 answers, but we have to factor it first using our cube formula. So X ^3 + 8 is going to be X + 2 * X ^2 -, 2, X plus 4 equaling 0. This first one's just going to give us X + 2 = 0 or X equal -2. The second one we're going to have to use quadratic formula or completing the square, so the opposite of -2 plus or minus the square root b ^2 - 4 * a * C all over 2A. So 2 plus or minus the square root 4 - 16 so -12 / 2. So 12 we could think of as 3 * 4. So we're going to get 2 square roots of three I all over two. We could think about factoring A2 out of the top, so we get 2 * 1 ± sqrt 3 I over 2, and then those twos would cancel, giving us 1 ± sqrt 3 * I. This next one we're going to multiply by the common denominator of X * X + 4, so we're going to get 7 * X + 4 + 7 * X equaling X * X + 4. Now recall X is never going to equal 0 or -4 because we can't have zero in the denominator. And I distribute everything out. We're going to combine like terms and we're going to take everything to one side so that we have 0 on the other. I personally like to have my squared terms positive, so I'm going to take the 14X and the 28 over. So X ^2 -, 10 X -28. So my a here is one. My B is -10 my C is -28 so X equals the opposite of B plus or minus the square root b ^2 -. 4 AC all over 2A so 10 ± 100 + 2 carry three eight 112 all over 2 so 10 ± 212. OK sqrt 212 two 12 is going to factor and a 21 O 625353 is prime. So we could think of 212 as there are two twos. So we can take sqrt 4 which is just two square roots of 53 / 2. If we factor out of two from the top, we get 5 plus or minus squared of 53 / 2. So then the twos cancelled leaving us 5 plus or minus squared of 53. Now you have a few for you to try. Thank you and have a wonderful day.