Olympiad Mathematics | How I got the Complete Solutions | Can You Solve This?

[00:00:00]Hi.

[00:00:03]Can I provide a complete solution to this equation here?

[00:00:11]We have a square root of a equals 8.

[00:00:17]Um how do we solve this problem here?

[00:00:22]Um because of this square root, you know, we have to remove it, right? To remove the square root here, we have to square everything on the left-hand side.

[00:00:33]And that means that 8 here will also be squared.

[00:00:38]So that the next step we're going to take is to say that what we have here is a squared multiplied by the square root of a squared because this can be written in this form.

[00:00:55]Then here we have 64 already.

[00:00:59]Now, a squared will come down multiplied by this and this will go, so we have just a.

[00:01:07]This is equal to 64.

[00:01:10]And if we take a step further, a multiplied by a is going to give us what?

[00:01:18]a to the power of 3, which is equal to 64.

[00:01:23]And the question says that we should solve this completely.

[00:01:27]So that means that at this point, we are going to get three solutions.

[00:01:33]So we have a to the power 3 to be equal to let's look at 4.

[00:01:39]4 is the same thing as Sorry, 64 is the same thing as 4 to the power of 3.

[00:01:46]Now, ordinarily, we could have said that let a be equal to 4 because if the powers are the same, then it means that the bases must be equal to each other.

[00:01:57]But if we do that from here, we're not going to have three solutions.

[00:02:02]So, we have to just bring this to the left. Now, a to the power of three minus four to the power of three is equal to zero.

[00:02:13]And from here, we understand that um we have difference of two cubes.

[00:02:20]Okay? And imagine that you have m cubed minus n cubed this is the same thing as m minus n then we multiply by m squared plus mn plus n squared.

[00:02:38]Okay, so this is what we we have.

[00:02:42]And um now our a will come here as m, so we have a minus n is four.

[00:02:52]Okay, so we open bracket m squared that's going to be a squared plus here we have mn which is going to be four n four a rather and we have our n squared which is 16.

[00:03:07]So, everything is equal to zero.

[00:03:11]And from here we apply zero product we apply zero product rule. Okay?

[00:03:23]And from the rule, we're going to say that a minus four is either zero or a squared plus four a plus 16 is zero.

[00:03:38]So, from this part already, our a is going to be equal to zero plus four and then we have four.

[00:03:46]So, at this point, we have a solution already and it is what? A equals 4.

[00:03:55]Now, to get the other solutions, we're going to bring this one down and then we'll solve it.

[00:04:03]Okay, so from here now, we're going to solve this one using quadratic formula.

[00:04:09]Oh, the formula is going to have a b c, right? So, let's say that um this equation is the same as x squared plus 4x plus 16 equals 0.

[00:04:22]So, that the formula we're going to use is minus b plus or minus, we have b squared minus 4ac and we divide by what? 2 times a.

[00:04:37]So, the next step is to get our values of a b c.

[00:04:41]A is the coefficient of x squared, which is 1.

[00:04:45]B is the coefficient of x, which is 4.

[00:04:49]And c is the constant, which is 16.

[00:04:53]So, let's put these three into this formula already.

[00:04:58]So, that our x will be equal to minus 4 plus or minus the square root of b squared which is going to be 4 squared minus 4 times a times c.

[00:05:13]A is 1, c is 16, right?

[00:05:17]So, from here Okay, we still divide by 2 times 1, right?

[00:05:23]Because a is still 1.

[00:05:26]Now, from here we are going to do this. X is equal to minus 4 plus or minus the square root of 16 minus 4 times 16.

[00:05:40]4 squared is 16 and is divided by 2.

[00:05:44]Now, I would have I could have multiplied this to get 64, but there's something else I want to do over there. So, we have this.

[00:05:53]16 is common to these two, so 16 is a common factor.

[00:05:58]Here we have 1 minus 4 will remain over there.

[00:06:03]Okay, so that's we divide all of these by two.

[00:06:08]Now, from here our X is equal to minus 4 plus or minus the square root of 16 multiplied by minus 3.

[00:06:20]1 minus 4 is minus 3, and this is over 2.

[00:06:24]So, if we go on from here, see what we can do.

[00:06:28]We know that the square root of XY can be written as square root of X times the square root of Y. You can split it this way.

[00:06:41]So, I'm going to split what we have on the other side.

[00:06:47]So, that X will be equal to minus 4 plus or minus we have the square root of 16 times the square root of minus 3.

[00:06:58]Now, this square root of minus 3, I can split it again to get square root of 3 times the square root of minus 1.

[00:07:07]So, that everything is still over 2.

[00:07:11]To continue from here our X is going to be minus 4 plus or minus we have square root of 16, which is 4.

[00:07:22]Square root of negative 1 is I, so multiply by 4.

[00:07:27]We'll have 4i.

[00:07:28]Then multiply everything here by root 3.

[00:07:31]This is because 3 is not perfect, so just to appear under the root.

[00:07:37]Now, we are dividing by 2.

[00:07:39]Now, let's do something here very, very quickly.

[00:07:43]The value of X will be 2 into minus 4 is minus 2 plus or minus 2 into 4i that is 2i and we have root 3.

[00:07:55]So, these are you know, this is a two-in-one solution, right? Because of the plus or minus.

[00:08:03]So, we have three solutions already.

[00:08:06]Let's bring them together very quickly.

[00:08:10]Okay, so this is the equation that we solved and we got A to be 4.

[00:08:17]Right? And from the last part, we had X.

[00:08:21]Remember that X was representing the A.

[00:08:24]So, A is equal to 2 negative 2 plus 2i root 3.

[00:08:32]Then, our our third solution will be A being equal to minus 2 minus 2i root 3. So, these are the three solutions to the equation.

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