The German Math olympiad problem that 99% of Students Get Wrong | Math Olympiad Mathematics

[00:00:00]Hello everyone. You're welcome to solve this nice square root math problem. This is the square root of a plus square root of b. This is equal to 35. Let's call this as equation one. Then we have the square root of a + b. This is equal to 25. We can call this as equation two.

[00:00:20]Now the question is what is the value of a? What is the value of b given that a and b these are element of real numbers.

[00:00:29]Now let's provide a solution from here from equation two. From equation two which is the square root of a + b this is equal to 25. The first step to do here let square on both sides.

[00:00:50]And this means that here we eliminate the square root sign. Therefore we have a + b. This is equal to 25 squar. This is 25 * 25.

[00:01:05]And therefore we have a + b. This is equal to 625.

[00:01:13]We can call this as equation three.

[00:01:16]Let's call this as equation three. Now from equation one, equation one is square roo<unk> of a plus square roo<unk> of b.

[00:01:29]This is equal to 35.

[00:01:33]The first step to do here let's square on both sides.

[00:01:39]So that now we have that square root of a + roo<unk> of b ^ 2. This is in the form of x + y raised ^ of 2 which we can express as x ^ of 2 + y ^ 2 + 2 x y.

[00:02:00]Applying this algebraic identity then here we have square roo<unk> of a raised ^ of 2 plus square roo<unk> of b raised to the power of 2 then + 2 multiplying by square root of a * square root of b. This is equal to 35 raised ^ of of 2.

[00:02:27]Now let's eliminate the square root sign here. This is a plus again here we have b then + 2 multiplying by square root of a * square root of b. We can express this as an individual square root of a * b.

[00:02:47]This is equal to 35 squared and this is 12 25.

[00:02:56]Now [clears throat] we have that a + b this is from equation three. Equation 3 which is = 625.

[00:03:06]Let's substitute 625 here. We have 625 + 2 *<unk> ab= 12 25.

[00:03:19]Now let's take 625 on the right hand side. So that we have to multiply by square root of a this is equal to 12 25 subtract 6 25.

[00:03:34]So we have 2 *<unk> a this is = 12 25 - 6 25 and this is equal to 600.

[00:03:47]So let's divide on both sides by two.

[00:03:52]And what we have here is that square root of a b this is equal to 600 / 2.

[00:04:00]This is equal to 300. This is equal to 300.

[00:04:05]So we have square root of a this is equal to 300. Now to solve for a let's square on both sides.

[00:04:14]And this means that a * b this is equal to 90,000.

[00:04:23]Let's call this as equation four. Let's call this as equation four. Now from equation three, from equation 3 which is a + b equal to 625.

[00:04:45]Let's square on both sides from here.

[00:04:49]So that now a + b ^ 2 this is the same thing as a 2 + b 2 + 2 * a this is equal to 625 raised to the power of 2.

[00:05:05]So we have that a 2 + b 2 + 2 * a is equation 4 and this is equal to 90,000.

[00:05:18]So let's substitute 90,000 here. This is equal to 625 squared and this is equal to 390,000 625.

[00:05:34]So this implies that we have a 2 + b 2 + 90,000 * 2 this is 180,000.

[00:05:44]This is equal to 390,625.

[00:05:51]Let's take 1 + 198,000 on the right hand side. So that we have a 2 + b 2 this is equal to so we have a² + b² this is equal to 390,625 - 180,000.

[00:06:13]And therefore we have a 2 + b² this is equal to we have 390,625 - 190 this is 210, 625. We can call this as equation.

[00:06:31]Equation five. Let's call this as equation five. In the next step we have that x subract y to the^ of 2. This can be expressed as x ^ 2 + y ^ 2 subtract 2 x y. This is an algebraic identity. And therefore, if we have a - b raised ^ of 2. This is the same thing as a 2 + b 2 subtract 2 * a b.

[00:07:09]Now this means that a subtract bis^ 2 this is equal to a² + b² this is equation 5 as you can see here. So we have 210, 625 subtract 2 multip is equation four and this is the same thing as 90,000.

[00:07:39]Now we have a - b ^ of 2. This is equal to 10,000 625 subtract 180,000.

[00:07:53]So this is to mean that a subtract b^ of 2 this is equal to 210,625 - 18,000 and this is equal to 30, 625.

[00:08:10]To solve for the value of a minus b, let's introduce a square root on both sides. We have the square root of a minus b raised to the power of 2. This is equal to plus or minus the square root of 30,625.

[00:08:28]Now let's eliminate the square root on both sides so that we have a minus b.

[00:08:34]This is equal to plus or minus the square root of that,625.

[00:08:41]This is the same thing as the square root of 175 raised to the power of 2.

[00:08:48]And let's eliminate the square root sign here. So that we have a minus b. This is equal to plus or minus 175.

[00:08:58]So we have two possible cases here. We have a minus b. This is = 175 [snorts] and we have a minus b this is equal to 175.

[00:09:13]Now given that a equation three we have a + b this is = 625 and we have a subtract b this is = 175.

[00:09:28]Again here we have a + b this is equal to 625.

[00:09:33]and we have a minus b this is equal to 175.

[00:09:40]Now from the first set of these are two set of re equations.

[00:09:45]Let's sum these two equations here. a + a this is 2 a + b + - b. This cancels out and this is equal to 625 + 175. This is equal to 800.

[00:10:03]Now let's divide both sides by two.

[00:10:07]And this means that a is equal to 400.

[00:10:11]So we have the value of a in case one is equal to 400.

[00:10:18]Let's solve for b.

[00:10:20]B is equal to 625 subtract a. So we have that b is = 625 subtract a which is 400.

[00:10:35]And what we have here we have that B is equal to 625 - 400 and therefore we have 225.

[00:10:47]We have the value of B is 225.

[00:10:50]So therefore in the first case that is in case one a comma b this is equal to 400 comma 225.

[00:11:06]This is the first set of solution. This is the first set of solution. In case two here in case two we have two set of frame equations.

[00:11:20]Let's sum these two set of linear equation here. A + A this is 2 A.

[00:11:28]We have B + - B. This cancels out 625 + - 175. So this is the same thing as 450.

[00:11:42]Let's simplify here. Let's divide both sides by two.

[00:11:46]What we have here is that a is equal to 225.

[00:11:53]A is = 225.

[00:11:58]Let's solve for B. B is = 625 subtract A. So that we have B is equal to 625 subtract A which is 225.

[00:12:13]Therefore we have the value of B in case 2 = to 400.

[00:12:19]In case two the value of B is 400.

[00:12:23]Now the second set of solution here for A, B.

[00:12:29]This is equal to 225, 400.

[00:12:35]This is the second set of solution that we have.

[00:12:39]So kindly follow the steps, like this video and kindly subscribe.

[00:12:45]See you in the next video and thank you for watching.