Can You Solve This in 10 Seconds? | Math Olympiad Problem

Added: 2026-05-23

[00:00:01]Hello, you're welcome. I just saw this nice exponential equation.

[00:00:06]What we have here is a 4 raised to power 6 to the left-hand side, we have x minus 2 raised to power 6 minus 4 raised to power 6 equals to 0 here.

[00:00:20]Which also can be written as x minus 2 raised to power 3 and all raised to power 2 minus also here we have 4 raised to power 3 raised to power 2 equals to 0.

[00:00:37]And this follows on we have a squared minus b squared which is the same thing as a plus b into brackets open bracket a minus b.

[00:00:48]Also here, a standing as x minus 2 raised to power 3 and b standing as 4 raised to power 3.

[00:00:59]I can write this here.

[00:01:02]And we have x minus 2 raised to power 3 plus 4 raised to power 3 into brackets also open bracket x minus 2 raised to power 3 minus 4 raised to power 3 close brackets equals to 0 here.

[00:01:23]Then we have two possible cases here.

[00:01:26]The first one x minus 2 raised to power 3 plus 4 raised to power 3 equals to 0.

[00:01:35]Or we have x minus 2 raised to power 3 minus 4 raised to power 3 equals to 0 here.

[00:01:44]Solving on this side, this follows on we have a raised to power 3 plus b raised to power 3 which can be written as a plus b into brackets open brackets a squared minus ab plus b squared close brackets then here I write this as x minus 2 plus 4 into brackets open brackets x minus 2 all squared minus 4 times x minus 2 plus 4 squared close brackets equals to 0 here.

[00:02:24]Then this becomes x plus 2 into brackets open brackets expansion here give us x squared minus 4x plus 4 then minus 4 open this bracket we have minus 4x plus 8 then plus 16 close brackets equals to 0 here.

[00:02:45]Then we have x plus 2 into brackets open bracket x squared minus 4x minus 4x give us minus 8x and 4 plus 16 20 plus 8 that's plus 28 close brackets equals to 0 here. We have two possible cases as well. Either this x plus 2 equals to 0 or we have x squared minus 8x plus 28 equals to 0 here.

[00:03:21]Solving on this side we have x equals to minus 2 which is a real solution here. Here we have a quadratic equation where a equals to 1 b equals to minus 8 and c equals to 28.

[00:03:38]Applying the quadratic formula which is x equals to minus b plus or minus square root of b squared minus 4ac all over 2a.

[00:03:51]Here we have x equals to minus minus 8 plus or minus square root of minus 8 square minus 4 times 1 times 28 over 2 times 1.

[00:04:12]So, this here we have x equals to minus times minus that's plus. We have 8 plus or minus square root of minus 8 square that's 64 minus 4 times 1 times 28 minus 112 over 2 here.

[00:04:34]So, this this becomes x equals to 8 plus or minus square root of 64 minus 112 that give us minus 48 over 2.

[00:04:49]Which also we can write as x equals to 8 plus or minus square root of 16 times 3 times minus 1 over 2.

[00:05:04]When we separate this here root 16 that's 4. We take it out. That is x equals to 8 plus or minus 4 and we have root 3 and root minus 1 is i over 2.

[00:05:18]Which implies we can factor 2 up here and we have x equals to 2 into bracket this remain 4 plus or minus even 4 [snorts] and we factor. But for the essence of what we want, so we have 4 plus or minus 2 root 3 i over 2.

[00:05:41]So, that is 2 cancel each other here.

[00:05:44]Then x now equals to 4 plus or minus 2 root 3 I.

[00:05:51]We have two complex solutions here.

[00:05:57]Solving from the second case here, this is also follows when we have a raised to power 3 minus b raised to power 3 is the same thing as a minus b into brackets, also open brackets, a squared plus ab plus b squared.

[00:06:15]Yeah.

[00:06:16]What we have becomes x minus 2 minus 4 into brackets, open brackets, x minus 2 all squared plus 4 times x minus 2 plus 4 squared close brackets equals to 0 here. That is here, we have x minus 6 into brackets open bracket x minus 2 here becomes x squared minus 4x plus 4. Then 4 open this bracket we have plus 4x minus 8 plus 16 close brackets equals to 0 here.

[00:06:59]This becomes x minus 6 into brackets, open brackets x squared minus 4x plus 4x this cancels out.

[00:07:09]We have 4 plus 16 that's 20. 20 minus 8 we have plus 12 close brackets equals to 0 here also.

[00:07:23]We have two possible cases, x minus 6 equals to 0 or x squared plus 12 equals to 0.

[00:07:34]On this side, this gives us x equals to 6, which is also a real solution here.

[00:07:41]Here we have a quadratic equation and when we take 12 to the other side, it becomes minus. You can see x squared equals to minus 12.

[00:07:49]Minus 12, yeah. [snorts] Then we take the square root on both sides. Square root of x squared equals to square root of minus 12.

[00:08:02]Square cancels roots.

[00:08:04]And this give us x equals to plus or minus square root of All right, this is 4 * 3 * minus 1.

[00:08:14]Now we separate this roots 4 as 2. We have x equals to plus or minus 2 root 3.

[00:08:24]And root minus 1 is i. So we have this as two complex solutions here.

[00:08:29]And therefore, altogether in this problem we have six solutions here, two real solutions here, two complex solutions here, and two complex solutions here.

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