Simplify the following expression: \(\frac{2}{(x+1)+\sqrt{x}}\)
Understand the Problem
The question asks to simplify the expression (\frac{2}{(x+1)+\sqrt{x}}). This likely involves rationalizing the denominator to remove the square root.
Answer
$$ \frac{2(x+1-\sqrt{x})}{x^2+x+1} $$
Answer for screen readers
$$ \frac{2(x+1-\sqrt{x})}{x^2+x+1} $$
Steps to Solve
- Rationalize the denominator Multiply both the numerator and the denominator by the conjugate of the denominator, which is $(x+1) - \sqrt{x}$.
$$ \frac{2}{(x+1)+\sqrt{x}} \cdot \frac{(x+1)-\sqrt{x}}{(x+1)-\sqrt{x}} $$
- Simplify the numerator Multiply the numerator:
$$ 2((x+1)-\sqrt{x}) = 2(x+1-\sqrt{x}) $$
- Simplify the denominator Multiply the denominator using the difference of squares formula, $(a+b)(a-b) = a^2 - b^2$, where $a = (x+1)$ and $b = \sqrt{x}$:
$$ ((x+1)+\sqrt{x})((x+1)-\sqrt{x}) = (x+1)^2 - (\sqrt{x})^2 $$
Expand $(x+1)^2$:
$$ (x+1)^2 = x^2 + 2x + 1 $$
So the denominator becomes:
$$ x^2 + 2x + 1 - x = x^2 + x + 1 $$
- Combine the simplified numerator and denominator The expression now becomes:
$$ \frac{2(x+1-\sqrt{x})}{x^2+x+1} $$
$$ \frac{2(x+1-\sqrt{x})}{x^2+x+1} $$
More Information
The simplified expression is $\frac{2(x+1-\sqrt{x})}{x^2+x+1}$. We removed the square root from the denominator by rationalizing it.
Tips
A common mistake is to incorrectly apply the difference of squares in the denominator, or to make an error when expanding $(x+1)^2$. Another common mistake is not distributing the 2 correctly in the numerator.
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