Only 1% Solve this Viral Math Olympiad | Can You Solve This One?

[00:00:00]In today's video, we solve for the value of a and the value of b given that a and b these are element of positive inteious. The problem is a + b + 4. This is equal to 4 multiplying by the square root of a multiplying by square root of b. Now let's provide a solution here.

[00:00:25]So we have a + b + 4. This is equal to now here we have 4 multiplying by we can express this as the square root of a multiplying by now we have that b is under 2 square root sign. So this means we have square t square roo<unk> of b.

[00:00:55]And therefore if we have the square root of b, this is the same thing as p raised to the power of a half.

[00:01:06]We have a square root of b. This is the same thing as b to the power of a half.

[00:01:10]Let's apply this property. So that now here we have a + b + 4 this is equal to 4 multiplying by a raised to the power of a half then multiplying by b raised to the power of a half then raised to the power of a half.

[00:01:35]So therefore we have a + b + 4 this is = 4 * a raised ^ of half then * b raised to ^ of 1 / 4.

[00:01:52]In the next step we have a + b + 4. This is equal to 4.

[00:02:00]multiplying by a ^2. This is the same thing as square roo<unk> of a multiplying by b ^ 1 / 4. This is the fourth root of b. This is the fourth root of b. In the next step, we can let square root of a b = x and we can let the fourth root of b be = y. We have that x is greater than z as well as y is greater than zero.

[00:02:36]So this is to mean that this is to mean that a is = x^ 2 and we have that y I mean b is = y raised to the power of 4. Now this means we raise to the power of 4 here. So that we have b is = y ^ of 4.

[00:02:59]Now let's go back to the equation which is a + b + 4. This is equal to the fourth root. I mean this is 4 multiplying by square root of a multiplying by the fourth root of b.

[00:03:17]So let's express in terms of x and y. So let us express this equation in terms of x and y. So we have that a is x². So we have x^ 2 + b. b is y ^ of 4 + 4. This is = 4 * square roo<unk> of a which is x * 4th root of b which is y.

[00:03:46]So we have x ^ of 2 + y ^ 4 + 4. This is = 4 xy. This is = 4 x y. So let's take 4 x y on the left hand side. So we have x² + y ^ 4 we have + 4 then subtract 4 xy this is = to 0. This is equal to z.

[00:04:17]Now let's rearrange this equation here.

[00:04:20]We have x^2 - 4 x y + y ^ 4 + 4. This is equal to 2 0.

[00:04:33]Now in the next step we have x ^ 2 subtract 4 x y then + y ^ 4. Now let's introduce + 4 y 2. This is + 4 y^ 2 subtract 4 y^ 2 + 4. This is = to 0. 4 y^ 2 - 4 y². This is the same thing as zero. So we have not out this equation here. And therefore we have x ^ 2 - 4 x y. Okay. Then plus let's add 4 y ^ 2. Then plus y ^ of 4.

[00:05:18]Then we have + we can subtract 4 y^ 2 then + 4. This is equal to to zero. This is equal to zero.

[00:05:32]Okay. In the next step we have x ^ of 2 subtract 4xy. This is the same thing as 2 * x then multip by 2 y + 4 y^ 2. This is the same thing as 2 y.

[00:05:54]This is ^ of 2. Then plus y ^ 4. This is the same thing as y^ 2. Then raised to the power of 2. Subtract 4 y^ 2. This is 2 y^ 2. then plus 4 we can express 4 as 2 raised to ^ 2 and this is equal to 2 0.

[00:06:19]Now in the next step we have that this is in the form of a² we can express in terms of m^ 2 - 2 m n + n² which is the same thing as m - n raised ^ of 2.

[00:06:44]So let's apply this algebraic identity in the two cases here. So therefore we have to we have x in the first instance here we have x subtract 2 y raised to the power of 2. Then we have plus in the second case here we have y ^ of 2 subtract 2 raised to the power of 2.

[00:07:13]This is equal to zero.

[00:07:16]This is equal to zero.

[00:07:18]So this means we have two instances here or two cases here which is x - 2 y raised to the power of 2. This is = 0.

[00:07:30]And we have y ^ of 2 subtract 2 ^ of 2.

[00:07:35]This is equal to 2 0.

[00:07:38]Now let's introduce square root of both sides. We have the square root of x - 2 y to ^ of 2. This is equal to the roo<unk> of 0. Here we have the square roo<unk> of y^ 2 - 2^ 2. This is equal to the square root of 0.

[00:08:03]Now let's eliminate the square root sign here. So that we have x - 2 y. This is = 0.

[00:08:14]And in that case we have x is = 2 y. We have x is = to 2 y.

[00:08:24]In case two here, let's also eliminate the square root side so that we have y^ 2 - 2. This is = 0.

[00:08:35]Let's take -2 on the right hand side.

[00:08:38]And this means y^ 2 this is = to two.

[00:08:44]Let's introduce square root on both sides. We have the square root of y 2 this is equal to plus or minus the square root of 2.

[00:08:56]So therefore we have so let's eliminate this square root sign here. We have y is = + or minus square<unk> of 2.

[00:09:08]Something to note here we have two possible values of y. y is = <unk>2 and we have y is = <unk>2.

[00:09:20]But something to note here we have that y is greater than zero. y is greater than zero. So therefore the second value of y here which is <unk>2 this is rejected because this is less than this is less than zero. So this is rejected.

[00:09:44]So we have the value of y in this case this is equal to square roo<unk> of 2.

[00:09:52]Now if you recall now we have x = 2 y and y is = roo<unk> of 2. Now if you recall we enter that the square root of we said that we let square root of a b = x and we end to let the square I mean the fourth root of b be = y. So in this case we have the fourth root of b. This is equal to y. y is square roo<unk> of 2.

[00:10:31]So what do we do next? Let's raise to the power of 4 on both sides.

[00:10:38]So that now here we eliminate the fourth root sign and therefore we have b is equal to we can express <unk>2 this is 2 ^ of 4.

[00:10:53]So therefore we have b is equal to now this is 2 ^ of a half * 4. So let's simplify here 4 / 2 this is 2. So we have b is = 2^ 2 which is 4.

[00:11:10]We have the value of b is = 4. Now let's solve the corresponding value of a. We have that x is = 2 y and given that y is = roo<unk> of 2 then x is equal to multiplying by square roo<unk> of 2. So this is the value of x this is the value of x and given that square root of a this is equal to x is 2<unk>2.

[00:11:44]So let square on both sides from here and this means that we eliminate the square root sign and therefore a is = 2^ 2 which is 4 *<unk> 2 raised to ^ of 2 and therefore a is = 4 * by by 2 and this is equal to 8.

[00:12:13]So we have the value of a is equal to 8.

[00:12:18]So we have that a comma b this is equal to 8 4.

[00:12:29]So this is the solution to this algebra problem. This is the solution to this algebra problem. So let's verify if this solution satisfies the equation.

[00:12:43]Now let's verify that a and b satisfies the equation. Now from here let's substitute a which is 8 + 4 + 4. This should be equal to this is 4 * square roo<unk> of a which is equal to 8 multiplying by square root of b which is<unk> 4.

[00:13:09]So we have 8 + 4 this is 12 + 4 this is 16.

[00:13:14]This should be equal to we have 4 * square roo<unk> of 8 *<unk> 4 which is 2.

[00:13:25]So we have 16 this should be equal to this is 4 *<unk> of 8 * 2 which is the square root of 16.

[00:13:37]So therefore we have 16 this should be equal to 4 * roo<unk> 16 which is 4. So we have 16 this is equal to 4 * 4 which is 16. In other words, we have the left add side is equal to the right add side and this affirms that the value of a and b which is equal to 8a 4 satisfies the equation.

[00:14:10]So kindly follow the steps like this video. If you have alternative solution method kindly showcase in the comment section. See you in the next video.

[00:14:22]Bye-bye for now.