A Nice Algebra Equation | Solve For "k" | Math Olympiad Mathematics

[00:00:00]K squared minus K raised to the power 3 is equal to 12. What is the value of K where K is a member of real number solution?

[00:00:14]K squared minus K raised to power 3 is equal to 12.

[00:00:21]Then we can rearrange this. We have minus K raised to power 3 plus K squared is equal to 12.

[00:00:34]Let's multiply all through by minus 1.

[00:00:37]Then here is K raised to power 3 minus K squared is equal to minus 12.

[00:00:47]Then K raised to power 3 minus K squared plus 12 is equal to zero.

[00:01:01]K raised to power 3 minus K squared plus 12 can be written as 8 plus 4 is equal to zero.

[00:01:13]Then K raised to power 3 minus K squared plus 8 can be written as 2 raised to power 3 plus 4 is 2 squared is equal to zero.

[00:01:28]Now let's compare K raised to power 3 plus 2 raised to power 3 minus K squared plus 2 squared is equal to zero.

[00:01:46]Here is K raised to power 3 plus 2 raised to power 3 minus K squared minus 2 squared is equal to zero.

[00:02:00]If you have a raised to the power three plus b raised to the power three, this is equal to a plus b multiplied by a squared minus ab plus b squared. Also, if you have a squared minus b squared, this is equal to a plus b multiplied by a minus b. Then, for this let's substitute.

[00:02:40]A is k plus two bracket k squared minus two k plus four minus Here is k plus two multiplied by k minus two.

[00:03:11]is equal to zero.

[00:03:14]>> [snorts] >> Then, k plus two bracket k squared minus two k plus four minus k minus two equal to zero. Because here, k plus two is common and k plus two is common, that's why we bring it out here. Then, here is k plus two bracket k squared minus 2 k plus 4 minus k minus times minus 2, that is plus 2 is equal to zero.

[00:04:06]k plus 2 bracket k squared minus 2 k minus k minus 3 k 4 plus 2 plus 6 is equal to zero. There are two cases here. The first case is k plus 2 is equal to zero.

[00:04:35]Therefore k is equal to minus 2.

[00:04:39]>> [snorts] >> Or the second case k squared minus 3 k plus 6 is equal to zero.

[00:04:49]This is a quadratic equation. So, which can be solved using quadratic general formula. And our quadratic general formula in this case is given as k is equal to minus b plus or minus square root of b squared minus 4 a c over 2 a.

[00:05:12]And from this equation a is equal to 1 b is equal to minus 3 c is equal to 6. [snorts] Therefore k is equal to minus bracket minus 3 plus or minus square root of minus 3 squared minus 4 multiplied by one multiplied by six over two multiplied by one or discriminant is equal to b squared minus four ac then b is minus three squared minus four multiplied by a is one and c is six then this is equal to minus three squared nine minus four times one times six twenty four equal to minus fifteen and this is less than zero so this is rejected because no real solution here therefore the only real solution we have is k equal to minus two this is the final answer we can check to confirm if our answer is right or wrong let's check we [snorts] have the given equation k squared minus k raised to power three is equal to twelve when k is equal to minus two here is minus two squared minus bracket minus two raised to power three must be equal to twelve then minus two squared is four minus bracket minus eight must be equal to twelve here is four minus times minus that is plus plus eight must be equal to 12. Then 4 + 8, this is 12 is equal to 12 and this satisfy the equation at K is equal to minus 2.