Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. A quadratic equation is an equation that we can put in the form AX squared plus BX plus C = 0. Once an equation is in this form, we can use the quadratic formula which says X equals the opposite of B plus or minus the square root b ^2 -, 4 AC all over 2A. Now the discriminates going to actually tell us what type of solution we have. If the discriminant equals 0, then this inside piece the discriminates b ^2 -, 4 AC. This inside piece would be 0, so we'd end up with negative b + sqrt 0 / 2 A and negative b -, sqrt 0 / 2 A. Or we get negative b / 2 a twice. So this is going to be 1 double root because it occurs twice and that double root is a rational number. So 1 double root rational solution. If the discriminates less than 0, we get sqrt b ^2 -, 4 AC. But that b ^2 - 4 AC is negative if it's less than 0. If it's less than 0, we know square root of a negative is I. So this is actually going to be two imaginary numbers. If the discriminate is greater than 0, then one of two things are going to happen. It's going to be a real number, but it's going to be either irrational or rational. So if b ^2 -, 4 AC is a perfect square sqrt 4, sqrt 9, sqrt 16, sqrt 25, we're going to get 2 rational roots. Otherwise, if it's not a perfect square, IE sqrt 3, sqrt 2, sqrt 7, it's going to be two irrational roots. Now both irrational and rational are real roots. So the first type of problem just says what type of solution. So all we care about is the b ^2 - 4 AC so -7 ^2 - 4 * 1 * 549 - 2029. So this is going to be two irrational roots because 29 was positive but it wasn't a perfect square. This next one b ^2 -, 4 AC so -8 ^2 -, 4 * a times C 64 + 2080. This one's 144. That is a perfect square. So this one's going to be two rational roots. If this number had become imaginary or if this number had been negative, it would have been two imaginary roots. And if this number had been zero, it would have been 1 double rational route. We've got a couple for you to try. The next type of problem is given solutions, we're going to find the equations. So if we know that X equal -5 and X equal 4, we could think about getting those set equal to 0. So X + 5 = 0 and X - 4 = 0. Now we're going to multiply the right sides together, and we're going to multiply the left sides together of those two equations. We're going to foil it out, so we get X ^2 + 5, X -4 X -20 zero times 00. If we combine like terms, we get X ^2 + X - 20 = 0. If we solve that by quadratic formula or factoring or completing the square, we would get the solutions -5 and four. This next 1X is going to equal 2/3 and X is going to equal -8. I actually don't like fractions, so I'm going to go ahead and multiply by three on each side and then I'm going to subtract the two. And here I'm going to add the 8 because I my goal is to get each of those equations equal to 0. And then we're going to multiply the left sides and we're going to multiply the right sides. When I foil this out, we're going to get three X ^2 + 24 X -2 X -16 and 0 * 0, just combining our like terms. Three X ^2 + 22 X -16 equals 0. The next example, same concept X = 7 I except it has imaginaries and it's OK, it'll work the exact same way. So X - 7 I equals zero X + 7 I equals 0. So when I foil the left sides and foil the right sides, I'm going to realize that the left side is actually the difference of two squares. So X ^2 + 7 nine minus 79 will cancel -49 I squared I ^2 is -1. So we're really going to end up with X ^2 - 49 * -1 and the negative times a negative is positive, so X ^2 + 49 = 0. This last example, X equal 3 plus root 2, X = 3 minus root 2. So X - 3 minus root 2 = 0 and X - 3 plus root 2 = 0. We're going to have a lot of distribution on this one. We're going to end up with 9 terms that we're going to then group together. So we're going to have X ^2 - 3 X plus squared of 2X minus three X + 9 - 3 root 2 - sqrt 2 X -3 sqrt 2 - sqrt 2 ^2. That all equals 0. So X ^2 - 6 X this positive root 2X and negative root 2X are going to cancel the negative root 3 two. Let's see -3 root 20. This one should have been a positive root 32. I lost my sign. Negative times up. Negative is a positive. Those will cancel. So we've got this +2 root X canceling with this -2 root X and this -3 root 2 cancel with this +3 root 2, and then the square root of a square cancels. So we really end up with plus 9 - 2 or X ^2 - 6 X +7 equaling 0. Thank you and have a wonderful day.