Olympiad Mathematics | Applying the addition of two cubes | Japanese

[00:00:00]Hi everyone.

[00:00:05]Can you provide the solution to this one here?

[00:00:11]Um, we have 2x to the^ of 3 + 8 to the power of 3 = 0.

[00:00:22]You know this is um very simple but we can decide to reduce this equation here because this can be 2 ^ 3 * what x to the power of 3. Then we have our + 8 to the power of 3. This is equal to zero.

[00:00:47]This is possible from one of the laws of indices.

[00:00:51]And now we can decide to divide all through by two to the power of three. So here I'm going to divide by 2 to power 3. This will be divided by 2 to power 3 and this will be divided by 2 to power 3 even though we're going to have zero on the right.

[00:01:12]Now remember that from one of the laws of indices um a to power 3 over b to the power 3 is the same thing as a over b of them to the power of three. Okay you have to take note of this right.

[00:01:34]So applying what I just talked about remember this one will go with this one first. So we have x ^ 3 + this will now be 8 / 2 to the power of 3 equals 0. 0 / 8 is going to be 0. So we are good. And um from here now x ^ 3 + 8 / 2 is 4. So that means we have 4 to^ 3 here. So this is equal to zero.

[00:02:09]And just like we always have um difference of two cubes, we're having addition of two cubes already. And if we have a cube + b cube, this addition is the same thing as a + b * a 2 - a + b 2. Okay, sorry I wrote out of sight.

[00:02:40]Okay, so this is what we have now. So that our a + b will now be x + 4. Close it into a squ is going to be x² - a b that's going to be 4x now right [snorts] then + b² our b² is going to be 4^ 2 and this is equal to zero.

[00:03:08]Okay. So from here what do we do? If we go on, we're going to get x + 4 in this bracket.

[00:03:18]And um in the next we're going to have x^2 - 4x + 16. Everything is equal to zero. So we can easily apply our zero product rule from here.

[00:03:36]Okay. So to apply our zero product rule because we have product of two terms here and it's producing zero. We will say that either x + 4 = 0 or x² - 4x + 16 is 0. And from this part our x is going to be 0 - 4. So the value of x is min -4 and this is our first solution.

[00:04:10]So to get the other solutions we have to bring this equation down and then solve it as a quadratic equation. What do we do?

[00:04:22]We understand that um it is a quadratic equation. So we're going to use the quadratic formula. So our a is 1. that is the coefficient of x² and then the b is going to be min -4 that is the coefficient of um of x then c is a constant which is 16.

[00:04:44]So the next step is to bring down the quadratic formula which is x = - b + or minus we have b^ 2 - 4 a c all over 2 a.

[00:05:01]So once you know your ABC you don't have to even look at the equation again. What you do is substitute the values of ABC into the formula given. And that's what I'm going to do. So that X will be - -4.

[00:05:22]Okay? Because B itself is -4 and C this negative has to come down to plus or minus. We have -4. There's a square on it. Then -4 * a cuz a is 1.

[00:05:37]then times C. Our C is 16.

[00:05:42]So we have to divide all of this by 2 * 1.

[00:05:47]2 * 1 will still give us um will still give us two, right? So from this point we have x to be 4 negative. Then we have - 4 2 is going to be 16. then minus 4 * 16 is going to be 64 and we are dividing all of this by 2.

[00:06:13]Now here is um one of the mistakes that some students make knowing that 16 and 64 are both perfect squares. They'll be tempted to find the square root of um both of them separately like now they're going to get 4 - 8 but that is not right. So what then do we do? we have to um subtract first before we carry out our work. So we have 4 + or minus 16 - 64 will give us - 48.

[00:06:50]Okay, it will give us - 48 and we divide by 2. So the next point let's um work work it here. we have x to be 4 plus or minus the square root of 48. You can see that I didn't put the negative right. So I have to bring the negative here which will appear under the root.

[00:07:13]So it is all over what? 2. But before then we can do something as well. 48 we know is the same thing as 16 * 3. I'm trying to reduce it. So we have this over 2. Now take another step as we have x to be equal to 4 + or minus square roo<unk> of 16 is 4 multiply it by the square root of -1 which is i. So that means you're getting 4 i over here then multiply by <unk>3.

[00:07:48]So this is all over what? Two and this implies that okay in fact let me say equal to two will go right two will go into four we have 2 plus or minus 2 will go into 4 I roo<unk>3 and we're going to have 2 I then roo<unk>3. So we have two solutions from here. Let's bring the complete solutions together.

[00:08:15]Okay. So this is the equation that we have solved and our first solution is -4.

[00:08:24]Our second solution is um 2 + 2 I <unk>3.

[00:08:32]And then the third one which is the last solution is 2 - 2 I <unk>3.

[00:08:41]So these are the three solutions to the equation.