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

追加: 2026-05-12

[00:00:01]In this video, let us solve for x given x * x * x + x = 10 x * x * x here can be written as x raised to power 3.

[00:00:29]then plus x is = 10.

[00:00:35]Let us bring this 10 to the left hand side. So we have x raised to power 3 + x - 10 is equal to z.

[00:00:51]then itself is equivalent to 8 + 2.

[00:01:01]So I'm going to replace 10 here with 8 + 2 to give us x^ 3 + x - 8 + 2 is = z.

[00:01:18]Then I'll open up this bracket. This will give us x^ 3 + x - 8 - 2 is = 0 and I'll write this as x^ 3 + x - 8 here is 2^ 3 then - 2 here is 2 raised to power 1 which I'll leave as two is equal to There I'm going to rearrange in order of power. I have x^ 3 - 2 rais^ 3 + x - 2 is equal to z.

[00:02:13]The first two terms gives us difference of two cubes. By identity given a raised power 3 - b raised power 3. We can factoriize this as a minus b into brackets a 2 + a * b then plus b 2.

[00:02:40]Using this to factoriize this, this will give us x - 2 into brackets x 2 + 2x + 2^ 2 then plus this. So we say + x - 2 is = 0.

[00:03:11]Let's tidy this up. This will give us x - 2 times all of this becomes x^2 + 2 x + 2 here is 4.

[00:03:26]Then plus I'll put this in brackets x - 2.

[00:03:32]So that we immediately see that x - 2 is common to these two terms. And we can factoriize that out to give us x - 2 into brackets x^2 + 2 x + 4 then + 1 = z giving us x - 2 into brackets.

[00:04:05]Here we have x^2 + 2 x 4 + 1 here is 5.

[00:04:14]All of that equal to zero.

[00:04:18]This would then imply that either x - 2 is = 0 or all of this x 2 + 2 x + 5 is = 0.

[00:04:38]From here we see x = 2.

[00:04:42]This is our first solution.

[00:04:45]We can get more solutions from this quadratic equation. I'll compare this with the general quadratic equation.

[00:04:52]ax^2 + bx + c = z. Then we see that a is here 1, b is 2 and the constant time c is 5.

[00:05:10]Then we use this formula x is = - b + or minus square<unk> of b^ 2 - 4 a c / 2 a.

[00:05:30]Since we have values for a, b, and c from here, we can then solve for x.

[00:05:37]Therefore, x is going to be minus b. b is 2. So, we have - 2 plus or minus the square root of b ^ 2, which is going to be 2 raised to power 2 - 4 * a is 1 and c is 5.

[00:06:03]All of that divided by 2 * a which will be 2 * 1 giving us x is = - 2 + or minus square root of 2 squ here is 4 - 4 * 1 is 4 4 * 5 is 20 square root of that / 2 * 1 is 2.

[00:06:36]So we have x is = -2 + square root of here we are going to have -6 then / 2.

[00:06:53]This will give us x is = - 2 + or minus. We can write this as square<unk> of 16 * -1 square root of that then / 2.

[00:07:13]So x is = -2 plus or minus we separate this into square roo<unk> of 16*<unk> -1 then / 2 which will give us x is = -2 plus or minus square root of 16 is 4 and roo<unk> of -1 is I all of that divided by 2.

[00:07:48]Now I can separate this division to give us X is = - 2 / 2 + or minus 4 I / two.

[00:08:06]Two here is one. Two here is one. Two here one. Two here is two. So we have x = -1 + or minus 2 i.

[00:08:21]Remember we got x1 earlier and the value we got for x1 is 2. And now in this x we just got there will be two values which we can split into x2 and x3. X2 is going to be -1 + 2 I and X2 X3 is going to be -1 - 2 I giving us all three possible values to this problem.

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