A Nice Algebra Olympiad Problem | How To Solve?

Added: 2026-05-09

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

[00:00:01]Welcome back to my channel, Hi I'm a mathematics. Today we have great, quick, and relaxing algebra question. We know that the sum a plus b equal to 16 and the product a times b equal to 20. So, we need to solve this question for a and b. So, if you have your solution, your answer, you can also write it in the comments below and then we will check your answer. So, write your solution and then we will check it.

[00:00:22]Okay, so first of all, what is the main idea? First of all, let's express from first equation, let's express b. This our first first step. So, every time you have system of equations, every time express something. In our case, let's express b.

[00:00:35]So, b equal to 16 minus a.

[00:00:38]16 minus a. And right now, let's plug in instead of this b, let's plug in 16 minus a into this second equation. So, let's do it right now. So, we have a times 16 minus a equal to equal to 20. Right now, let's simplify it. We have only one variable, we have only a, so it's it's really good.

[00:00:58]Let's multiply this a by this parenthesis. So, we have 16 a minus a square and let's write this 20 on the left side. Let's do it. So, we have minus 20 equal to equal to zero. And right now, we have looks like this a quadratic equation. Here we have square, we have a, we have constant, but we need to reorder this. So, we need to change this order real quick. First of all, let's write this minus a square on the first position. So, minus a square on the first position plus 16 a plus 16 a on the second position this coefficient and minus 20 minus 20 equal to zero.

[00:01:34]And the final tricky move.

[00:01:36]I recommend you multiply both side by minus one. A lot of students might be thinking that this is not necessary, but I I prefer solving this quadratic equation with the basic order of signs.

[00:01:47]So, that's why I don't need this negative sign right here and then I have a classic order. So, a square without negative sign with a positive and right here we need to change all the signs.

[00:01:57]So, minus 16 a plus 20 equal to zero. Okay, so this our first step. So, we really hope you understand how can we get this quadratic equation.

[00:02:08]What are we going to do next? Right now, let's solve this quadratic equation. Of course, there are a lot of ways how can we do this, but maybe the basic one is basic of method of coefficients. So, that's why let's write that coefficient a equal to one, our b equal to this our second coefficient equal to minus 16 minus 16 and our third coefficient c equal to equal to 20. And right now, let's solve this quadratic equation. First of all, I'm going to find I'm going to solve it for a discriminant. So, b square minus 4 a c equal to So, b square So, minus 16 square minus 4 times a times 1 and times c times 20. So, right now, let's simplify it. So, we plug in each of these elements into this into this spot. Let's do it.

[00:02:56]Uh so, what we going to do next? Minus 16 square equal to 256.

[00:03:01]This our table case minus 4 times 20 equal to equal to 80. So, equal to 176.

[00:03:09]So, the first new is that our discriminant is greater than zero. It's also really good. So, right here we have two uh two roots two roots two real number roots. If you want to uh say more, okay, two real number roots. So, right now, let's let's solve it for for this root. So, a first and a second. So, minus b this our classic formula plus minus square root of discriminant and all over all over 2 a.

[00:03:36]Okay, so minus b minus minus 16 minus 16.

[00:03:41]Plus minus square root of discriminant square root of 176 176 and all over uh 2 a 2 times 1.

[00:03:51]Okay, so let's simplify it. So, minus minus 16 equal to 16. So, 16 plus minus square root of 176. How can we express this 176? The best way to do it is to write it as 16 times 11. Okay, this equal to 100 76 and all over two. Why do I choose this 16 times 11 not like another way? Because square root of 16 equal to four and so this our table case. That's why I we express this square root of 176 as a product of two values.

[00:04:25]One of these should be like table case, yeah?

[00:04:28]So, we have 16 plus minus square root of 16. So, according to a square root property, we can split it like square root of 16 times square root of 11.

[00:04:37]So, a square root of a product equal to product of square roots over two.

[00:04:42]So, we have 16 plus minus 4 square root of 11 4 square root of 11 over over two. And our final step, let's factor our two from our from our numerator, yeah? So, let's let's plug in and let's factor it. So, two in in parenthesis, we have eight plus minus 2 square root of 11 2 square root of 11 and all over all over two. So, as a result, our answer is eight plus minus 2 square root of 11. This is our a first and a second. So, let's write it. So, a first with the positive sign right here with the positive sign equal to eight plus 2 square root of 11 and a second with a negative sign equal to eight minus 2 square root of square root of 11.

[00:05:35]This is not our solution.

[00:05:37]This only a first and a second, but in the beginning we had that our b equal to 16 minus a. So, that's why we need to find we need to solve it for b first and for b second. So, that's why b first equal to 16 minus a first. So, 16 minus a first and right here equal to 16 minus a second. So, let's plug in it. So, b first equal to 16 minus eight plus 2 square root of square root of 11. So, b first equal to 16 minus eight minus 2 square root of 11. So, b first from here equal to right here we have eight eight minus 2 square root of 11, yeah?

[00:06:24]And according to the same logic, b second equal to 16 minus a second eight minus 2 square root of 11. Let's simplify this real quick.

[00:06:34]So, b second equal to 16 minus eight plus 2 square root of 11. And our b second equal to eight plus 2 square root of 11. One really interesting pattern we can see, yeah?

[00:06:47]We have eight we have the same absolutely the same constant, but different signs. Here we have eight plus 2 square root of 11. We have plus right here we have minus. Right here we have minus. Right here we have plus. So, we have looks like symmetrical pairs. So, let's do it. Let's write our final answer and then we will check it. Okay, so our answer So, eight plus 2 square root of 11 and eight minus 2 square root of 11. This our first pair and the second pair eight minus 2 square root of 11 and eight plus 2 square root of 2 square root of 11.

[00:07:24]Okay, really great. And in the end, let's check real quick. Let's prove real quick our our roots. So, a plus b our system of equations looks like that. So, a plus b equal to 16 and a b equal to equal to 20. Okay, really great. And right now, let's check. One really interesting hint, one really interesting moment we have right here addition and multiplication. And a plus b equal to uh the same as b plus a. So, every time we have symmetrical roots. So, doesn't matter. We can we can check only one pair of roots. We don't need to check both of these because we will have the same answer because of this addition. So, we can change this a and b by places. So, a plus b equal to b plus a. We have symmetrical roots. So, it doesn't matter.

[00:08:08]In our case, let's check for example addition of these two elements.

[00:08:13]So, a plus b equal to eight plus 2 square root of 11 plus eight minus 2 square root of 11 equal to 16. Let's check it. So, it's it's great because we can right here we can cancel it. So, eight plus eight equal to 16. So, addition is addition is great. And of course, you can also check this one, but we will have absolutely the same thing. You can easily see it without this without this thing right here. Eight minus this one eight plus this one. So, we can cancel this. We will have absolutely the same thing.

[00:08:46]And multiplication is absolutely the same thing as addition because a times b equal to b times a. So, that's why we don't need to check both of these because we can change places right here.

[00:08:57]So, we can we don't need to check both of these. We will have absolutely the same root because of multiplication. It works perfectly with addition and multiplication, not with subtraction and division, yeah?

[00:09:07]So, a times b equal to eight plus 2 square root of 11 and eight minus 2 square root of 11. And a lot of students might be thinking, okay, we need to multiply parenthesis by parenthesis, but just look at this closely. Eight plus something eight minus something. So, a x plus y x minus y. We can easily change it by x plus y. So, we need in our head, yeah? x plus y and x minus y.

[00:09:35]And then it And then it like a formula from school our classic identity a x square minus y square.

[00:09:42]So, difference of squares. Yeah, it is.

[00:09:44]This our formula from this our school formula x square minus y square. So, difference of squares. First element square eight square minus 2 square root of 11 square. We raise this to the second power. And right now, 64 minus What do we have right here? Uh two squares, we need to raise this two to the power two and this one. So, right here we have four, two square four, and square root of 11 to the power two, we're going to cancel the square root sign, so we have only 11. 4 * 11 equal to 40 44. So, equal to 20. And we have this A * B equal to 20, so the expression that we exactly need.

[00:10:24]So, our root is absolutely great. This is our answer, so I don't have enough space, so that's why I'm going to leave our answer right here. So, we have two pairs of of roots. This is the first one, this is the second one, both are correct.

[00:10:39]So, thank you for your time.

[00:10:40]Have a great day. Also, write your thoughts, write your respond in the comments below. It will be really interesting to read about it. Write your Write your notes, write your question, what do you think about this type of question in the comments below. It will be really interesting to read about it.

[00:10:52]We wish you all the best in life. Take care of yourself. Have a great day. See you in the next videos.