Germany | Can you solve this ? | Math Olympiad | X=? & Y=?

Added: 2026-05-25

[00:00:00]Hello. You're welcome to solve this math problem, which is x + y is equal to 24.

[00:00:08]x * y is equal to 64.

[00:00:11]To find the values of x and y's from these two systems of equations.

[00:00:16]Now, in the first step, let's start by letting this here as equation one and this as equation two.

[00:00:24]Then, from equation one, which is this, x c plus y is equal to 24.

[00:00:33]Into here, we make y the subject, so we take x to the left side. So, it will be y is equal to 24. Take x to this side, it will be minus x.

[00:00:44]Then, note this equation in terms of y.

[00:00:48]Then, in the next step, from equation two, what is equation two? It is x c * y is equal to 64.

[00:00:57]Now, into y, we substitute 24 minus x.

[00:01:01]So, it will be x c * y, which we substitute here, which is 24 minus x.

[00:01:09]Bracket is equal to 64.

[00:01:13]Then, here it will be x * 24, it is 24 x.

[00:01:18]x * negative x is negative x c squared is equal to 64.

[00:01:25]Then, we start by this negative x c squared. Then, plus 24 x c. We take 64 to this side, it will be minus 64 is equal to zero.

[00:01:40]Then, into here, we make this negative x c squared, we make this positive. So, then we divide the whole equation divide by negative one.

[00:01:51]So, it will be negative x c squared divide by negative one is positive x c squared.

[00:01:57]Then, 24 x c divide by negative is minus 24 x.

[00:02:04]-64 / -1 is positive 64 is equal to 0 / -1 it is 0.

[00:02:14]Then from this quadratic equation, we solve this by using quadratic formula.

[00:02:19]Now by applying quadratic formula to find the values of x is equal to -b plus or minus square root of b squared minus 4ac over 2a.

[00:02:34]So here it will be x is equal to minus b is coefficient of x which is -24.

[00:02:43]Then plus or minus square root of b squared it will be -24 bracket squared.

[00:02:52]Then minus 4 * a a it is 1 * c c is 64.

[00:03:01]Then over 2 * a it is 1.

[00:03:07]Then into here it will be x is equal to negative negative negative and negative 24 is positive 24 plus or minus square root of -24 squared it will be negative squared is positive. Now 24 squared is 576.

[00:03:27]Then minus 4 * 4 it is 16 with 1 4 * 6 is 24. 24 plus 1 is 25.

[00:03:39]Then over this and this is 2.

[00:03:43]Then into here it will be x is equal to 24 plus or minus square root of we take this minus this Now, 6 - 6 is 0. 7 - 5 it is 2.

[00:03:57]5 - 2 it is 3. Then over 2.

[00:04:03]So, into here it will be x is equal to 24 plus or minus square root of 320.

[00:04:12]Now, here it will be square root. Let's find the prime factors of 320.

[00:04:17]Now, from 320 here divide by 2 160.

[00:04:23]Then divide by 2 80. Then divide by 2 40.

[00:04:30]Then divide by 2 20.

[00:04:33]Then divide by 2 10.

[00:04:37]Then divide by 2 5.

[00:04:40]Then divide by 5 1.

[00:04:43]Now, here it will be 2 * How many 2s? 1 2 3 4 5 6. So, here it is 2 ^ 6 then times this 5 over this 2.

[00:04:58]Then in the next step it will be x is equal to 24 plus or minus we separate the square root. So, it will be square root of 2 ^ 6 times square root of 5 over 2.

[00:05:14]Then here it will be x is equal to 24 plus or minus square root of 2 ^ 6 it is 2 ^ 3 then times this square root of 5 over 2.

[00:05:29]Then into here it will be x is equal to 24 plus or minus 2 ^ 3 it is 8 then times square root of 5 divide by 2. Here we divide by 2 and into this side divide by 2.

[00:05:46]So, here it will be x is equal to 24 / 2 it is 12 plus or minus 8 / 2 it is 4 square root of 5.

[00:05:58]So, from here we have two solutions.

[00:06:01]Whereas the first value of x is equal to when it is positive it is 12 plus 4 square root of 5.

[00:06:12]And the second value of x here is equal to when it is negative is 12 minus 4 square root of 5.

[00:06:24]Now, from here I've got the values of x.

[00:06:27]Now, to get y we apply this equation in terms of y. Y is equals 24 minus x.

[00:06:33]So, from y is equal to 24 minus x then here this is x1 so it will be y1 is equal to 24 minus x1 we substitute here 12 plus 4 square root of 5.

[00:06:53]So, here it will be y1 is equal to 24.

[00:06:57]Here we take negative to be minus 12 minus 4 square root of 5.

[00:07:04]So, here it will be y1 is equal to 24 minus 12 it is 12 minus 4 square root of 5.

[00:07:13]So, this is the value of x1 this is y1.

[00:07:18]Then to solve from this second solution this is x2.

[00:07:22]So, here it will be we apply y is equal to this here which is 24 minus x.

[00:07:30]So, x2 here it will be y2 is equal to 24 minus x2 it is 12 minus 4 square root of 5.

[00:07:42]So, here it will be y2 is equal to 24.

[00:07:46]We take negative inside, so it will be minus 12. It will be plus 4 square root of 5.

[00:07:54]Then here it will be y2 is equal to 24 minus 12 it is 12. Then here plus 4 square root of 5.

[00:08:05]So, this is value of x2. This is y2.

[00:08:10]So, our conclusion x1, y1 this is from the first solution.

[00:08:18]x1 it is this, y1 it is this. So, here it is 12 plus 4 square root of 5, y1 y1 it is 12 minus 4 square root of 5.

[00:08:35]Uh the second solution which is x2, y2 is equal to x2 it is this. Here this is y2. So, here it is 12 minus 4 square root of 5, y2 y2 it is 12 plus 4 square root of 5.

[00:08:59]So, these are all the values of x and y from this our problem.

[00:09:06]Now, in the next step let's check this answer if it is correct. So, to check from our problem which is x plus y is equal to 24 and we have x times y is equal to 64.

[00:09:26]So, x plus y it will be this plus this.

[00:09:30]So, x it is 12 plus 4 square root of 5 plus y it is this here from the first solution.

[00:09:40]12 - 4 square root of 5 is it equal to 24?

[00:09:48]Now, 4 square root of 5 - 4 square root of 5 is 0. So, this and this here it is 0. Then it will be 12 + 12 is 24 is equal to 24.

[00:10:00]So, left side and right side are equal.

[00:10:02]Then we already check by using the first equation.

[00:10:07]Now, let's check by using the second equation. x * y, so it will be this * this. So, x it is 12 + 4 square root of 5 bracket * y, it is this here, which is 12 - 4 square root of 5.

[00:10:29]Then is it equal to this 64?

[00:10:35]Now, here to multiply 12 * 12, it is 144.

[00:10:40]Then 12 * -4 square root of 5, it will be -12 * 4 is 48 square root of 5.

[00:10:49]Then 4 square root of 5 * 12, here it will be + 4 * 12 is 48 square root of 5.

[00:10:58]Then here 4 square root of 5 * -4 square root of 5, it will be -4 square root of 5 bracket square. Then is it equal to 64?

[00:11:12]Then here, 48 square root of 5 - 48 square root of 5 is 0.

[00:11:17]So, this and this will cancel, then it will be here 144, then 4 square it is 16.

[00:11:27]Square root of 5 square it is 5.

[00:11:31]Then is it equal to 64?

[00:11:35]Now here to be 144 minus 16 * 5 it is 80. Then is it equal to 64?

[00:11:45]Now here 4 minus 0 it is 4.

[00:11:49]14 minus 8 it is 6. Then is it equal to This is equal 64 is equal to 64. So left side and right side are equal.

[00:11:59]Then we need to check for by using this second equation.

[00:12:03]So this first solution here it is correct. Also the second solution is correct because the value of x in the first solution it is the value of y in the second solution.

[00:12:15]And the value of y from the first solution it is the value of x in the second solution. So they interchange the values.

[00:12:25]Thank you for watching.

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[00:12:34]Bye-bye.