A Nice Olympiad Algebra Problem | How to solve for 'a and b' in this problem ?

Added: 2026-05-13

[00:00:00]Hello. How to solve this math problem which is a² minus b is = 133.

[00:00:09]b² minus a is equal to 133. To find the value of a and b such that a is not equal to b. Now solution.

[00:00:25]In the first step, let's start by letting this here as equation one and this as equation two. Now here we'll take equation one minus equation two. So from the left side here it will be a² - b which is equation one. So here a² - b then minus from equation two it will be b square minus a. So here bracket b² - a bracket is equal to in the right side it will be 133 - 133 it is zero.

[00:01:08]Then into here it will be a² - b we take negative inside so it will be - b²us and minus it will be + a is = z.

[00:01:22]Then here it will be we start by this.

[00:01:25]So a² minus b² then + a - b is equal to zero.

[00:01:37]Now here a square - b square this here is same as a - bracket bracket a + b bracket then plus this here a - b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b = z.

[00:01:57]Now into this equation a minus b is common. So we'll take a minus b bracket outside the bracket. Now this here a minus b it is this a + b then plus a minus b / a minus b it is 1 bracket is equal to zero.

[00:02:22]Then from here we have two solutions where this first solution of a minus b is equal to zero and this second solution of a + b + 1 is = z. Now from this first solution we takeative b to the right side. So it will be a is = b.

[00:02:46]But note the condition for our problem a is not equal to b. So here a is not equal to b. So this solution here will be rejected. Then we continue to solve from this second solution. We take one to the right side. So it will be a + b is =1.

[00:03:06]So here let's call this equation three.

[00:03:11]Then in the next step in the first step we take equation one minus equation two.

[00:03:18]Now into here we'll take equation 1 + equation two.

[00:03:25]Now from equation one it is a² - b. So here a² - b + equation 2 it is b² - a. So here b² - a is equal to into here we'll take 133 + 133.

[00:03:51]So here 133 + 133.

[00:03:57]Then here it will be this plus this. So a² + b² then - a - b is equal to this plus this is 26.

[00:04:14]Then here it will be a² + b² is equal to this 266.

[00:04:25]Here minus a take to this side will be plus a minus b you take to this side it will be + b.

[00:04:33]Then in the next step it will be a² + b² is equal to 266 into here a + b is from equation 3 a + b is -1. So here it will be + a + b is -1.

[00:04:55]Then it will be a² + b² is = 266 - 1 it is 265.

[00:05:06]Now let's call this equation 4.

[00:05:12]Then in the next step we recall from equation equation three. Whereas equation 3 it is this a a + b is equal to -1. So from a + b is = -1 into here we square both sides. So it will be a + b bracket² is = -1 bracket squared. Now the expansion of this it is a² + b² + 2 a b is = -1² it is 1. Then from a square + b square is from equation 4 which is 265.

[00:06:02]So from a square + b square we substute 265.

[00:06:08]Then + 2 a b is = 1.

[00:06:14]Then here it will be we take 265 to this side. So it will be 2 a b is equal to 1.

[00:06:22]This side to be - 265.

[00:06:28]Then here it will be 2 a b is equal to 100 - 265 it is - 264.

[00:06:40]Then here we divide by two in both sides also into here divide by two. So this and this will cancel. Then it will be a b is equal to this / by this it is minus 132.

[00:06:57]Then let's call this equation five.

[00:07:02]Now into here we have we have a + b. Now let's find a minus b.

[00:07:10]So we recall so we recall the rule of expansion of a - bracket² whereas this is equal to a² + b² - 2 a b. Now into here it will be a - bracket square is equal to a square + b² is from equation 4. Whereas a square + b square it is this 265 then - 2 * a it is this here. So bracket -32 bracket then into here it will be a - bracket² is equal to 265 negative and negative it will be positive 2 * 2 it is 4 2 * 3 is 6 2 * 2 it is 2 then here it will be a - bracket 8² is equal to we add here this plus this it is 9 6 + 6 is 12 go with 1 2 + 2 is 4 + 1 it is 5 then here to remove this square we'll apply square root in both sides so it will be square root of a - bracket square is = square root of 529 so into Here this square root you can say square then it will be a - b is equal to square root of this here it is plus or minus 23.

[00:09:06]Then into here or you can check this square root of 529 here by two it is 2 * 2 is 4. Then here minus you left one. Then we drop two and 9. 2 * 2 it is four. Then here * 3 here * 3. 3 * 3 it is 9. 3 * 4 it is 12.

[00:09:33]So here square root of this number it is 23 plus or minus. So into here we have two solutions of a minus b. So we have a - b is equal to when it is positive it is 23 and a - b is equal to when it is negative it is - 23.

[00:09:59]Then here you have a minus b. Then we recall from equation three whereas we have a + b is -1.

[00:10:09]So from a + b is = -1. Also here we apply the same equation a + b is = -1.

[00:10:24]Now here let's solve this continuous equation by elimination method. So here we will add these two equations. Then a + a it is 2 ab + b is 0 is equal to 23 +1 it is 22.

[00:10:44]Then here divide by 2 in both sides. So this and this will cancel then it will be a is = 22 / 2 it is 11.

[00:10:56]Now to get the value of b here we will apply this rule here of a + b is = -1 here then we substitute a is 11. So it will be 11 + b is = -1.

[00:11:16]Then here it will be b is = -1. We take 11 to this side it will be -1.

[00:11:25]Then here it will be b is = -1 - 11 it is -2.

[00:11:33]So here got the value of a and this is the value of b.

[00:11:40]Then in the next step here let's solve this second equation here second solution. So we'll add these two solutions. So it will be a + a it is 2 a - b + b is 0 is equal to -23 + -1 is -4.

[00:12:02]Then here divide by 2 in both sides. So this and this will cancel then it will be a is equal to -4 / 2 is -12.

[00:12:14]Then here we apply this equation here of a + b is = -1.

[00:12:25]Then from a we substitute this here -12.

[00:12:29]So here it will be -12 + b is = -1.

[00:12:35]Then here it will be b is = -1. -12 * this side it will be + 12.

[00:12:44]Then into here it will be b is = 12 - 1 it is 11. So here we have the value of a and this we have the value of b. Now our conclusion a comma b from the first solution a comma b is equal to a it is 11 b is -12.

[00:13:11]So here it will be 11, -12.

[00:13:15]And from the second solution, a is -12, b is 11. So here a comma b is equal to -12, 11. So these are all the values of a and b from this our problem. Now in the next step let's check this our answers if they are correct. So to check from our problem whereas our problem it is a² - b is =33.

[00:13:57]Also we have the other equation b² - a = 133.

[00:14:05]Now here let's check for this first solution here of 11, -12.

[00:14:13]Now by using this first here first equation of a² - b is =33 we will use a comma b is = 11 -12.

[00:14:32]So here it will be we substitute a is 11. So it will be 11² - b is -12. So here -12 is 8 = 133.

[00:14:47]So here it will be 11² it is 121.

[00:14:51]Negative negative to be positive 12 is it = 133.

[00:14:59]Now this plus this is 133 is equal to 133. So left side and right side are equal. Then it is true for this solution.

[00:15:12]Now in the next step let's check for this second solution here of a comma b is equal to where they interchange the values here b it is a so -12 a is ba 11 same as here -12 11 now here let's check we will check only for by using this first equation a square - square= to this. So a square - b square I mean a squareus b is equal to 133.

[00:15:53]So we substitute our answers. So it will be -2 bracket² - b is 11.

[00:16:03]So it will be - 11 is it = 133.

[00:16:08]So - 12² it will be positive 144 - 11 is it = 133.

[00:16:18]Now this minus this is 133 is equal to 133. So left side and right side are equal then it also true for the second solution.

[00:16:31]You can check by using this this here this answer here this equation in the second the second equation from this problem. But all the answers they are correct because they interchange their values.

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[00:16:57]Bye-bye.