Can you solve this cube root equation? | Algebra Challenge

Hinzugefügt: 2026-05-31

[00:00:00]Hello and welcome. In this math tutorial, our task is to find all the real values of x that satisfy this cube root equation. Now, to solve this problem, let us start with two substitutions.

[00:00:13]Let a be equal to the cube root of 6 + root x, while b is equal to the cube root of 6 root x. Now, of course, you know that if we raise both sides of this first equation to power three, then we have that a cubed is equal to 6 + root x. And in the same way, when we raise both sides of this second equation to power three, then we have that b cubed is equal to 6 root x.

[00:00:54]Now, notice that if we add these two equations, we will be able to eliminate x.

[00:01:01]On the left-hand side, we have a cubed + b cubed. And on the right-hand side, we have 6 + 6, which is equal to 12. Of course, root x - root x is equal to zero.

[00:01:16]Now, from the original equation, it is easy to see that a + b is equal to the cube root of three. So, we have a + b equal to the cube root of three.

[00:01:30]The next thing we are going to do is to raise both sides of this second equation to power three.

[00:01:37]Now, when we expand the left-hand side, we have a cubed + 3 a squared b + 3 a b squared + b cubed. And of course, on the right-hand side, the cube is going to take care of the cube root, leaving us with three.

[00:02:02]Now, let us rearrange the terms on the left-hand side. We have a cube plus b cube plus When you look at these two terms, you'll see a common factor, which is 3ab.

[00:02:19]So, we have 3ab into 3a²b / 3ab is going to give us a.

[00:02:27]And 3ab² / 3ab is going to give us b. Now, this is still equal to three.

[00:02:36]But now, remember that a cube plus b cube is equal to 12. So, this is 12, and a plus b is equal to the cube root of three.

[00:02:49]So, we can write this equation as 12 plus 3 * the cube root of three * ab is equal to three.

[00:03:05]Now, let us subtract 12 from both sides of this equation.

[00:03:09]When we do that, we have that 3 * the cube root of three * ab is equal to three minus 12.

[00:03:20]3 - 12 is equal to minus nine.

[00:03:25]Now, the next thing we are going to do is to divide both sides of this equation by 3 * the cube root of three.

[00:03:36]This is going to cancel that, so that on the left-hand side of this equation, we have ab, and this is equal to minus nine over 3 * the cube root of three. Now, three into nine is three. So, this is equal to minus three over the cube root of three.

[00:04:01]Now, we can simplify this using the laws of indices. This is equal to minus three to the power one divided by three to the power one over three.

[00:04:19]We have minus three to the power one minus one over three. Of course, you know that when we divide two numbers with the same base, all we have to do is to subtract their exponents. Now, three times one is three. Three minus one is two. So, this is equal to minus three to the power two over three.

[00:04:44]Now, at this point, remember that this is A and this is B.

[00:04:50]So, when we multiply these two, we have the cube root of six plus root X times six minus root X.

[00:05:04]And this is equal to minus three to the power two over three.

[00:05:09]Now, to get rid of this cube root, we raise both sides of this equation to power three. When we do that, on the left-hand side, we have six plus root X times six minus root X. And on the right-hand side, we have minus three to the power two over three raised to power three.

[00:05:31]Now, of course, you know that this is the same as minus one raised to the power three.

[00:05:38]times three to the power two over three raised to the power three.

[00:05:46]Minus one raised to the power 3 is -1.

[00:05:50]This will cancel that. 3 squared is 9.

[00:05:54]So, we have that 6 + √x * 6 √x is equal to -9.

[00:06:07]Now, you know that the product of these two factors is going to give us the difference of two squares because they are conjugates. 6 squared is 36.

[00:06:18]√x squared is x. This is equal to -9.

[00:06:25]Rearranging this equation, we have that 36 + 9 is equal to x. Now, 36 + 9 is equal to 45. So, that simply means that the only real value of x that satisfies this cubic equation is x equal to 40 5.

[00:06:50]And with that, we come to the end of this tutorial. I hope you learned something. If you enjoy such content, please subscribe to the channel. Give us a like to support the channel. Thanks for watching, and you can see more tutorials here.