Simplify the following expression: 2 / (x+1 + sqrt(x))
Understand the Problem
The question is asking to simplify the expression 2 / (x+1 + sqrt(x)). This likely involves rationalizing the denominator by multiplying by the conjugate.
Answer
$\frac{2x + 2 - 2\sqrt{x}}{x^2 + x + 1}$
Answer for screen readers
$\frac{2x + 2 - 2\sqrt{x}}{x^2 + x + 1}$
Steps to Solve
- Identify the conjugate
The conjugate of the denominator $(x+1) + \sqrt{x}$ is $(x+1) - \sqrt{x}$.
- Multiply the numerator and denominator by the conjugate
Multiply both the numerator and denominator by $(x+1) - \sqrt{x}$: $$ \frac{2}{(x+1)+\sqrt{x}} \cdot \frac{(x+1)-\sqrt{x}}{(x+1)-\sqrt{x}} $$
- Simplify the numerator
The numerator becomes $2((x+1) - \sqrt{x}) = 2(x+1 - \sqrt{x}) = 2x + 2 - 2\sqrt{x}$.
- Simplify the denominator
The denominator becomes $((x+1) + \sqrt{x})((x+1) - \sqrt{x})$. This is a difference of squares, so it simplifies to $(x+1)^2 - (\sqrt{x})^2 = (x^2 + 2x + 1) - x = x^2 + x + 1$.
- Write the simplified expression
The simplified expression is $\frac{2x + 2 - 2\sqrt{x}}{x^2 + x + 1}$.
$\frac{2x + 2 - 2\sqrt{x}}{x^2 + x + 1}$
More Information
Rationalizing the denominator is a common technique used to simplify expressions, particularly when the denominator contains a radical. It involves multiplying both the numerator and denominator by the conjugate of the denominator to eliminate the radical from the denominator.
Tips
A common mistake is not correctly identifying the conjugate or making an error when expanding and simplifying the denominator after multiplying by the conjugate. It's important to remember the difference of squares formula: $(a+b)(a-b) = a^2 - b^2$.
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