German Math Olympiad | Can you solve for a,b?

Hinzugefügt: 2026-05-30

[00:00:00]Hello everyone. Happy to see you here.

[00:00:01]Today, let's solve this really interesting algebra question. We need to solve this question for A and B. So, we know that the sum A + B equal to 10 and the [clears throat] product A * B equal to 24. We need to solve it for A and B.

[00:00:14]So, if you have your solution, your assumption, you can also write it in the comments below and then we will check your answers. So, it will be really interesting to test our algebra skills.

[00:00:22]Okay, so first of all, let's express from the first equation, let's express our B. Let's do it. So, our B from the first equation equal to 10 - A. 10 - A.

[00:00:34]And right now, let's plug in instead of this B, let's plug in this 10 - A to the second equation. So, let's do it. So, we have A * instead of B we plug in 10 - A.

[00:00:45]So, 10 - A. And equal to 24. 24.

[00:00:51]Right now, let's simplify it. So, we we can multiply this A by this parentheses.

[00:00:56]So, we have 10 A A squared and equal to 24.

[00:01:02]Right now, let's change an order because we can clearly see that this is a quadratic equation, but we prefer our - A squared on the first position. The next with A, we have + 10 A.

[00:01:14]+ 10 A and we have - 24. - 24 equal to 0.

[00:01:21]Right now, let's multiply both side by by -1. Okay? Multiply both side by -1.

[00:01:29]And we can change all the signs to the opposite one. A squared - 10 A.

[00:01:35]And + 24. So, right now we can easily say, "Okay, we have very great order of signs because we prefer this A squared with the positive sign." So, we have a classic quadratic equation. So, what we're going to do next? Of course, we're going to solve it, but let's first of all, my quick recommendation, let's write our coefficients. So, our A equal to 1.

[00:01:56]B equal to -10 and C equal to 24. Okay, we have this coefficients. And right now, let's solve for discriminant real quick. So, let's find our discriminant.

[00:02:07]So, B squared - 4 A C equal to B squared. So, -10 squared.

[00:02:14]-10 squared - 4 A C - 4 * 1 and * >> [clears throat] >> 24.

[00:02:21]All right. Right now, let's simplify this. -10 squared equal to 100.

[00:02:29]4 * 24 equal to 96. So, -96 equal to 4.

[00:02:35]And this is extremely great discriminant for us because this discriminant is positive, so it implies that in this branch we have two two roots. And moreover, we have two real number roots.

[00:02:46]For example, if we have because sometimes we have discriminant which is less than 0. So, it means that we have two two complex two complex roots. But in our case, we have two discriminant positive discriminant, so we have two real number roots. So, let's solve for this real number roots. We have A first and A second equal to -B. This is our all known formula plus minus square root of discriminant and all over 2 A. So, let's plug in. We know everything. We know discriminant, we know B, we know A and and C. So, first of all, -B. So, - -10.

[00:03:26]Plus minus square root of discriminant.

[00:03:29]Square root of 4.

[00:03:31]And all over 2 A. 2 * 1.

[00:03:36]Okay, minus minus 10 equal to 10 plus minus square root of 4 equal to 2. Over over 2. So, basically we have two branches. 10 + 2 over 2 and 10 - 2 over 2. So, let's write for example, our A first. A first equal to So, we have 10 + 2 over 2 equal to What do we have? 12 over 2 equal to 6.

[00:04:01]And A second. A second equal to 10 - 2 over 2. So, equal to I guess this is equal to 4 because we have 8 over 2 equal to 4. So, right now we have A first equal to 6. This is our A first.

[00:04:19]And we have A second. A second equal to equal to 4. This is not a solution to this question. This is not a 6 and 4.

[00:04:27]This is this is not a pair because we need to solve it for B as well because B equal to 10 A. So, basically we need to solve this one for B first.

[00:04:39]Equal to So, right now let's solve it.

[00:04:42]So, B first equal to 10 - A first. And you can clearly see that this is equal to 10 - 6. So, B first equal to equal to 4. So, we have our first pair.

[00:04:56]B first equal to 4 and A first 6 and B first 4. Right now, let's solve it for B second. B second equal to 10 - A second. So, B second equal to 10 - 4.

[00:05:10]Equal to 6. So, B second equal to 6. And we have our pairs 6 and 4 and 4 and 6. So, let's write it as a final answer and then we will check our our roots. So, let's do it. So, our answer to this question we have first pair with the first index.

[00:05:29]So, we have 6 and 4.

[00:05:33]And 4 and 6. Right now, let's check our roots real quick. In the end, let's check it. So, we have A + B equal to 10.

[00:05:42]And AB equal to 24. I really hope you see that addition works perfectly. And one really interesting moment, we don't need to check both of these parentheses.

[00:05:51]We need to check only, for example, one of these because in terms of addition, A + B is the same as B + A. So, 6 + 4 is the same as 4 + 6. So, 6 + 4 is the same as 4 + 6 is equal to 10. So, addition works perfectly.

[00:06:07]>> [clears throat] >> Right now, let's check real quick our our multiplication. So, right here we have multiplication.

[00:06:13]A * B. So, we have 6 * 4. And moreover, a multiplication we can also change positions. We can write this not like A * B, but also B * A. So, that's why we don't need to check both of these, but maybe sometimes you have like complicated roots, so you need to check both of these. But in our case, we need to check only one of these. 6 * 4.

[00:06:33]Moreover, we can write 4 * 6. This is not a mistake, but we don't need to do this because this is the same 24. So, our roots are absolutely absolutely correct. Of course, in terms of in terms of mathematics, in terms of if you're talking about school, you can easily guess this root by inspection because every time you have this type of system of equation A + B equal to 10, AB equal to 24 and a lot of students they have this system of equation. So, A + B equal to 10 and AB equal to equal to 24. And you know, a lot of students, maybe 90% of students, they might be thinking, "Okay, let's try to solve this question by inspection." And they say, "Okay, 6 and 4. 6 + 4 10. 6 * 4 24." So, yeah, they say, "Okay, 6 and 4." This is a good thing, but you forget about another roots. 4 and 6. This is also really really popular mistake because a lot of students solve this question by inspection. And I don't want to say that this is correct because this is absolutely wrong. If you solve this question by inspection, you first of all, you skip a solution to this question because Why Why do we learn math? Because we need to know how can we solve this question.

[00:07:45]Yeah, we forget about this solution. And secondly, if you solve this question by inspection because it happened sometimes that the students want to solve this question by inspection, but secondly, you forget about another roots.

[00:07:58]So, basically with this inspection method, you skip a solution. This is the first thing. You skip a solution. And secondly, you skip other possible roots because you you're not sure that 6 and 4 is only one one root. You forget great thing because a lot of students want to solve this question in an easy way. And my quick recommendation, every time you have this type of question, and no matter you know a solution or you don't know, try to solve this question.

[00:08:26]And if you're talking about this one, especially about this one, this is not a hard question. So, don't be scared about it. If you have this system of equation, like don't solve this question by inspection method. Try to do something, express something, plug in something, and then you might be thinking, "Okay, I have a quadratic equation, but I know how to solve a quadratic equation."

[00:08:43]Yeah, and then discriminant and we have A first and second. And then you you might be thinking, "Okay, I'm really proud of of myself that I solved this question." We have a our two two roots.

[00:08:55]And I have like not one one inspect one pair but which is by inspection. Yeah?

[00:09:02]We have We have the second pair as well.

[00:09:04]You know, this is like a few thoughts from me from a from a teacher's perspective because this is extremely great thing to mention it because a lot of students skip this part. And just my quick recommendation, just forget about inspection method. For example, you sit in your exam and you have like 1 minute left. So, maybe in that way you can easily write something like that. But this is very bad very bad expression when you write like a question and an answer. This is extremely weird for you and of course and for your teacher. My quick recommendation, try to solve this question. Especially about this one, this is not a hard question. We are talking about discriminant. We are talking about basic mathematical thing, basic mathematical step, and we have a classic classic answer. So, it was my thought about this question. I really hope you understand it. I really hope you understand what I mean because I don't know is exactly I don't know how many students, how many teachers is watching this video right now? Because maybe a lot of students and I want to say this extremely important thing. If you're you're a teacher, I hope you you understand what I mean. A lot of a lot of students, a lot of different things happen at school, yeah? And a lot of different ways, a lot of different situation, especially in mathematics.

[00:10:16]So, thank you everyone for watching.

[00:10:17]Wish you all the best in life. Take care of yourself. Also, write your thoughts, write your response in the comments below. I want to say thank you everyone for for being here because my main goal is to make math clear and understandable for everyone. And if this video is helpful, I'm extremely happy that you you understand something new, you learn you learn something new. And of course, this is extremely great a great thing.

[00:10:38]So, thank you everyone for your time.

[00:10:39]Take care of yourself. Have a great day and see you in the next videos.