Math Problem Statement

x222x1\frac{x^2-2}{2x-1}

Solution

The expression given is:

x222x1\frac{x^2 - 2}{2x - 1}

Simplifying the Expression

We aim to simplify this rational expression if possible.

  1. Factorize the numerator and denominator if applicable:

    • The numerator, x22x^2 - 2, cannot be factored further as it does not have rational roots.
    • The denominator, 2x12x - 1, is already in its simplest form.
  2. Check for common factors: Since there are no common factors between the numerator and the denominator, the expression cannot be simplified further.

Restrictions on xx:

To ensure the denominator is not zero: 2x1=0    x=122x - 1 = 0 \implies x = \frac{1}{2}

Thus, x12x \neq \frac{1}{2}.

Final Answer:

The simplified form of the expression is: x222x1,where x12.\frac{x^2 - 2}{2x - 1}, \quad \text{where } x \neq \frac{1}{2}.

Would you like to analyze this further or evaluate it for specific values of xx?


Related Questions:

  1. How would you find the asymptotes of this rational function?
  2. Can you evaluate this expression for x=2x = 2?
  3. What is the domain of this function?
  4. How can this function be graphed?
  5. What happens as xx \to \infty or xx \to -\infty?

Tip:

Always check for restrictions on the denominator before performing any operations with a rational expression.