Germany | Can you solve this? | A tricky German math olympiad question |

[00:00:00]Hello everyone. Welcome to Russell's Classroom. Today we are solving a interesting math question which is x + y is equal to 16, xy is equal to 16, and x, y is equal to what?

[00:00:14]How to solve this interesting math question?

[00:00:20]So I solve this question easy method.

[00:00:21]This is our first equation and this is our second equation.

[00:00:27]So first of all, I take our first equation.

[00:00:30]So here our first equation which is x plus y is equal to 16. This is our first equation.

[00:00:39]Now at this moment, if I move on this x in this side, it will be negative. So you can say here y is equal to 16 minus x. I just move on this x in this side.

[00:00:51]So we are find out y is equal to 16 minus x. This is our third equation.

[00:00:56]Now at this moment, I take our second equation because of that here xy is equal to 16. This is our second equation.

[00:01:06]>> [snorts] >> So this is x and times this y. Why it will be 16 minus x? I substitute this value here, so it will be 16 minus x and this is 16.

[00:01:20]Now at this moment, we are find out a nice quadratic equation because of that 16 times x, it will be 16x.

[00:01:27]Then x times x, this is x to the power two and this is 16.

[00:01:34]Now this exponential equation which is minus x squared then I take this value, it will be positive 16x.

[00:01:44]And if I move on this 16 in this side, it will be negative 16 is equal to zero.

[00:01:51]Then this quadratic equation, if I multiply both side by negative one or negative sign, this is positive, this is negative, this is positive. So, this is x to the power two and this is minus 16x and this is plus 16 is equal to zero. So, we have find out a nice quadratic equation. And at this moment, you can say here a is equal to one and b is equal to minus 16 and c is equal to plus 16.

[00:02:22]So, we have find out here is a is equal to one, b is equal to minus 16 and c is equal to 16. Now, I apply this math formula here which is x is equal to minus b plus minus square root, this is b squared minus four ac over two a.

[00:02:48]Then, here x it will be minus b.

[00:02:53]And b is minus 16. So, I take this value here, so it will be minus and b is equal to minus 16.

[00:03:00]Then, plus minus this is a square root b squared minus four ac. So, b squared it will be minus 16 squared and minus four ac. A is one and c is 16 over two a. A is one.

[00:03:20]Then this is x is equal to At this moment, you can say this is minus minus, it will be plus 16.

[00:03:29]And this is plus or minus square root.

[00:03:33]You can say 16 squared, it will be 256.

[00:03:37]And I can say this is positive value because of that power is even number. 16 * 16, here is minus minus, it will be plus. So, it will be 16 squared. Okay?

[00:03:47]Then, it will be minus four times 16 over two times 1 it will be 2.

[00:03:54]Then here x it will be here 16. Then you can say but it's plus minus square root.

[00:04:04]Here is 16 is common. So if I take 16 is common so this is divide this it will be 16 minus this divide 16 it will be four. So we have find out it will be four over two.

[00:04:21]Then here x it will be 16 then plus or minus square root look this is 16 okay but this is 12.

[00:04:35]This is 12. So 16 * 12 over two.

[00:04:41]Now this is x is equal to 16 plus minus. If I separate this case you can say it will be square root 16 it will be four and square root 12 I take this 12 here this 12 here over two.

[00:05:02]Then here x is equal to 16 plus minus and four and square root 12 you can say it will be four times three. Four times three this is 12 over two. Then here x it will be 16 plus minus then four times square root four it will be two times and this is square root three it will be square root three over two.

[00:05:34]Then here x is equal to 16 plus minus then four times two this is eight and here is square root three over two. Now, if I separate this fraction, we are find out X is equal to 16 divide two plus minus four. Sorry, this is eight. If I divide by two, it will be four.

[00:06:02]This is eight square root three over two.

[00:06:08]Now, this is X is equal to eight plus minus four square root three.

[00:06:17]And I I mean divide this 16 divide two, it will be eight. This eight divide two, this is four. So, we are find out eight plus minus four square root three. This is the value of X.

[00:06:27]This is the value of X in this math question. But, Y is equal to what?

[00:06:31]So, you can say here X one, it will be eight plus four square root three.

[00:06:39]And X two it will be eight minus four square root three. Because of that, here is two solution.

[00:06:46]Uh first solution is plus, and second solution is negative.

[00:06:51]So, at this moment, Y is equal to what?

[00:06:53]So, remember that our third equation, which is Y is equal to 16 minus X. Our target Y is equal to what? So, I take our third equation.

[00:07:02]So, which is uh third equation which is Y is equal to 16 minus X. So, Y is equal to 16 minus X.

[00:07:18]So, we find out here is two solution.

[00:07:20]This is Y one, this is Y two. Okay. So, this is 16 minus X. Remember that X is equal to eight plus four square root three.

[00:07:32]And here 16 minus X. Remember that X is equal to eight minus four square root three.

[00:07:41]Then, this is 16 - 8 - - it will be minus plus minus this is - 4 square root 3. So, you can say 8 - 16 - 8 it will be 8 - 4 square root 3.

[00:07:57]And here 16 this is - 8 this is + 4 square root 3 minus minus this is plus so it will be 8 plus 4 square root 3. So, we are finding out here is our final solution which is x {comma} y when x is equal to 8 + 4 square root 3 8 + 4 square root 3 this time y is equal to 8 - 4 square root 3 both are both are positive solution both are real solution here is no complex number and here also 8 - 4 square root 3 this is 8 + 4 square root 3.

[00:08:42]Now, let's verify our question which is x + x + y is equal to 16. This is our question.

[00:08:55]And xy is equal to 16. This is our question.

[00:09:00]So, remember that here is x is equal to this y is equal to this. So, I take this value here. Okay?

[00:09:07]So, if I take both value here so x it will be 8 plus 4 square root 3.

[00:09:15]Here is plus so plus y it will be 8 - 4 square root 3. This this cancel 8 + 8 this is 16. So, you can say here is 16 so left hand side and right hand side both side is equal.

[00:09:31]But here xy is equal to 16. Is it right?

[00:09:33]Let's check out.

[00:09:35]Here is x is equal to 8 + 4 root 2 over 3. So I take this value here, it will be 8 plus 4 root 3. Then here is 8 minus 4 root 3. Now you can apply here is a plus b times a minus b, which is a squared minus b squared. I mean a squared minus b squared it will be a plus b times a minus b. Then this is 64 and this is 4 squared it will be 16. And a squared will be times and it will be 3 3 times 16, this is 48. Then 64 minus 48 it will be 16. So here also left hand side and right hand side both side is equal. So you can say easily x is equal to this and y is equal to this, x is equal to this and y is equal to this. This is our final answer in this tricky German math problem.

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[00:10:26]Goodbye. Take care everyone.