Solve for t in this nice Algebra equation | Math Olympiad Mathematics

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[00:00:01]In this video, let us solve for t given t raised to power 2 minus 2 raised to power 3 is equal to 12.

[00:00:21]We are given t raised to power 2 minus t raised to power 3 is equal to 12.

[00:00:31]Let us bring 12 to the left-hand side.

[00:00:33]This will give us t raised to power 2 minus t raised to power 3 minus 12 is equal to 0.

[00:00:44]Then we get t raised to power 2 minus t raised to power 3 then minus I'm going to split 12 here to give us 4 + 8.

[00:01:00]Then if we open up this bracket to get t raised to power 2 minus t raised to power 3 minus 4 minus 8 is equal to 0.

[00:01:15]We can also rewrite this as t raised to power 2 minus t raised to power 3 minus 4 here is 2 raised to power 2 then 8 is 2 raised to power 3.

[00:01:32]is equal to 0.

[00:01:35]I'm going to rearrange this by order of power so that we have t raised to power 2 minus 2 raised to power 2 then minus t raised to power 3 minus 2 raised to power 3 is equals to 0.

[00:02:00]This would then imply t raised to power 2 minus 2 raised to power 2 minus t raised to power 3 plus 2 raised to power 3 is equals to 0.

[00:02:22]From here we have addition of two cubes.

[00:02:25]But here we have difference of two cubes.

[00:02:30]By identity given a raised to power 2 minus b raised to power 2 we can factorize this to give us a minus b times a plus b.

[00:02:47]So this here becomes t minus 2 times t plus 2.

[00:02:58]Then minus t raised to power 3 plus 2 raised to power 3 is equals to 0.

[00:03:10]As mentioned earlier, this is addition of two cubes.

[00:03:14]By identity given a raised to power 3 plus b raised to power 3 we can factorize this as a plus b into brackets a squared minus ab plus b squared.

[00:03:33]So using these two factorize this, we're going to have t raised to power 3 plus 2 raised to power 3 is equals to t plus 2 into bracket t squared minus 2t.

[00:03:54]Then + 2 squared.

[00:03:58]So, we're going to put this in place of this here to give us t - 2 * t + 2 then minus t + 2 into bracket t squared - 2t. This will be a + 4.

[00:04:28]Then equal to 0.

[00:04:33]We have t + 2 here, t + 2 here. We can factorize that to give us t + 2 into bracket t - 2 then minus all of this t squared minus 2t + 4 then equal to 0.

[00:05:01]We get t + 2 into bracket t - 2 - t squared + 2t - 4 is equal to 0.

[00:05:20]t + 2 into brackets I have t here I have 2t here giving us 3t then minus 2 minus 4 that gives us minus 6 and then minus t squared is equal to 0 which will then imply that either t + 2 is equals to 0 or 3t - 6 - t squared is equals to 0.

[00:06:02]From the first equation, which is a linear equation, we get t is equals to -2.

[00:06:08]We'll call this t1.

[00:06:11]And then the second equation, which appear to be a quadratic equation, we can rewrite as -t squared + 3t -6 is equals to 0.

[00:06:29]We divide through by -1, this will give us t squared -3t +6 is equals to 0.

[00:06:40]Now, we solve this quadratic equation for two more values of t.

[00:06:46]To solve for t here, let's first compare with the general quadratic equation ax squared + bx + c equals to 0.

[00:06:58]So, that we get the values of a, b, and c.

[00:07:02]a is 1, b is -3.

[00:07:07]And c is 6.

[00:07:09]Then, we go ahead and solve for t using the general quadratic formula, which is -b + - square root of b squared -4ac then divided by 2a.

[00:07:30]We have our values for a, b, and c. So, let's plug into this formula.

[00:07:34]This will give us t is equals to -b which is -3 + square root of minus three squared minus four times one times six then divided by two A which is two times one.

[00:07:59]This will give us T is equal to positive three plus or minus square root of negative three squared here will give us nine minus four times one is four four times six is 24.

[00:08:17]Square root of that then divided by two times one is two.

[00:08:23]So this gives us T is equal to three plus or minus square root of nine minus 24 will give us negative 15.

[00:08:37]Then divided by two.

[00:08:41]So we have T is equal to three plus or minus square root of this will be 15 times negative one divided by two.

[00:08:59]If we separate the radical we get T is equal to three plus or minus square root of 15 and then times square root of negative one then divided by two.

[00:09:15]This will give us T is equal to three plus or minus This will be I square root of minus one is I then root 15.

[00:09:29]Divided by two.

[00:09:34]Let us separate this division here. This will give us T is equal to 3 over 2 plus or minus I root 15 divided by 2.

[00:09:50]And then we split this into two different values of T.

[00:09:53]Recall that earlier we got a value for T as T1 which gave us -2.

[00:10:03]And now here we are going to have T2 equal to 3 over 2 plus I root 15 divided by 2 and T3 equal to 3 over 2 minus I root 15 divided by 2.

[00:10:32]These are all possible solutions from this problem. We have two complex roots and only one real solution.

[00:10:52]The next thing we are going to do now will be to do a quick check with the real solution.

[00:10:58]The problem given was T raised to power 2 minus T raised to power 3 is equal to 12.

[00:11:07]And now we got T1 which is the only real solution to be -2.

[00:11:14]So, I'm going to put in place of T in this equation -2.

[00:11:21]This would then imply -2 raised to power 2 minus -2 raised to power 3 to give us 12.

[00:11:34]-2 raised to power 2 give us 4.

[00:11:38]- -2 raised to power 3 will give us -8.

[00:11:44]That will give us 12.

[00:11:46]This will give us 4 + 8 is equals to 12.

[00:11:53]4 + 8 is 12, so we have 12 is equal to 12.

[00:11:58]And since the left-hand side balances the right-hand side, that confirms that the solution we got for T here is absolutely correct.

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[00:12:14]And I'll see you in my next video.

[00:12:17]Bye.