2x^3 = x
Understand the Problem
The question is asking us to solve the equation 2x^3 = x for the variable x. This implies isolating x and understanding its possible values.
Answer
The solutions to the equation are $x = 0$, $x = \frac{\sqrt{2}}{2}$, and $x = -\frac{\sqrt{2}}{2}$.
Answer for screen readers
The solutions to the equation $2x^3 = x$ are:
$$ x = 0, \quad x = \frac{\sqrt{2}}{2}, \quad x = -\frac{\sqrt{2}}{2} $$
Steps to Solve
- Rearranging the equation
We start with the equation given, which is $2x^3 = x$. To solve for $x$, we need to move all terms to one side of the equation.
We can rewrite this as:
$$ 2x^3 - x = 0 $$
- Factoring the equation
Next, we can factor out the common term, which is $x$:
$$ x(2x^2 - 1) = 0 $$
- Setting each factor to zero
Now, we set each factor equal to zero to find the possible values for $x$.
First, we have:
$$ x = 0 $$
Next, we solve the second factor:
$$ 2x^2 - 1 = 0 $$
- Solving the quadratic equation
To solve for $x$ in the second factor, we can isolate $x^2$:
$$ 2x^2 = 1 $$
Dividing both sides by 2, we get:
$$ x^2 = \frac{1}{2} $$
Now, we take the square root of both sides:
$$ x = \pm \sqrt{\frac{1}{2}} $$
This simplifies to:
$$ x = \pm \frac{\sqrt{2}}{2} $$
- Summarizing the solutions
Putting all our solutions together, we find:
$$ x = 0, \quad x = \frac{\sqrt{2}}{2}, \quad x = -\frac{\sqrt{2}}{2} $$
The solutions to the equation $2x^3 = x$ are:
$$ x = 0, \quad x = \frac{\sqrt{2}}{2}, \quad x = -\frac{\sqrt{2}}{2} $$
More Information
The solutions found include one zero solution and two irrational solutions. The value $ \frac{\sqrt{2}}{2} $ is a common approximate value used in trigonometry as well.
Tips
- Not factoring the equation correctly; it's important to always look for common factors.
- Forgetting to include both positive and negative roots when taking square roots.
- Misplacing or omitting terms when rearranging the equation.
