- #1
anemone
Gold Member
MHB
POTW Director
- 3,883
- 115
Determine all real \(\displaystyle x\) satisfying the equation \(\displaystyle \sqrt[5]{x^3+2x}=\sqrt[3]{x^5-2x}\)
topsquark said:Just a note: Wolfram|Alpha will not give you correct solutions as written. It will give x = 0 as the only real solution. There are three real solutions. So no cheating!
-Dan
Actually that one ( and [tex]-\sqrt{2}[/tex]) aren't that hard to find. We know that there has to be more than x = 0, otherwise the problem is too simple. There are a number of 2's floating around there so it would seem wise to check solutions of the form [tex]2^{a/b}[/tex]. This yields the [tex]\pm \sqrt{2}[/tex] solutions, but really doesn't address the general problem.Bacterius said:Wolfram|Alpha also gives $x = \sqrt{2}$ as a real solution. One step closer to brute-forcing the problem! (Smoking)
I thought of that approach as well. The original equation is odd on both sides, so we know that any roots will be of the form y = x0, -x0. But note that when you raise the equation to the 15th power only the positive solutions survive.MarkFL said:I find that when I raise both sides to the 15th power, divide through by $x^3$
Hey, it's been a long couple of days... (Sleepy)MarkFL said:Are you certin raising the original equation to the 15th power loses negative roots? Recall, we still found $x=-\sqrt{2}$ after doing so.
A radical equation is an equation that contains a radical expression, such as a square root, cube root, or higher index root. The goal is to solve for the variable within the radical expression.
To solve a radical equation, you must first isolate the radical on one side of the equation. Then, you can raise both sides of the equation to the power that will eliminate the radical. Finally, solve for the variable using basic algebraic operations.
The solutions to a radical equation can be real or complex numbers. When solving for real solutions, it is important to check for extraneous solutions, which are values that make the original equation undefined.
Yes, there can be multiple solutions to a radical equation. This is because raising both sides of the equation to a certain power can produce multiple values that satisfy the equation. It is important to check all possible solutions and identify the correct ones.
Yes, there are specific steps to follow when solving a radical equation. These include isolating the radical, raising both sides to eliminate the radical, checking for extraneous solutions, and simplifying the solution. It is also important to check the solution in the original equation to ensure its validity.