Simplify the expression: 2 / (x+1 + √x)
Understand the Problem
The question is asking to simplify or rationalize the denominator of the given expression. This involves algebraic manipulation to remove the square root from the denominator by multiplying both the numerator and denominator 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 the denominator by the conjugate $(x+1) - \sqrt{x}$: $$ \frac{2}{(x+1)+\sqrt{x}} \cdot \frac{(x+1)-\sqrt{x}}{(x+1)-\sqrt{x}} $$
- Simplify the numerator
The numerator becomes $2((x+1) - \sqrt{x}) = 2(x+1) - 2\sqrt{x}$.
- Simplify the denominator
Using the difference of squares formula, $(a+b)(a-b) = a^2 - b^2$, the denominator becomes: $$ ((x+1)+\sqrt{x})((x+1)-\sqrt{x}) = (x+1)^2 - (\sqrt{x})^2 $$ $$ = (x^2 + 2x + 1) - x = x^2 + x + 1 $$
- Write the simplified expression
Combine the simplified numerator and denominator: $$ \frac{2(x+1) - 2\sqrt{x}}{x^2 + x + 1} = \frac{2x+2 - 2\sqrt{x}}{x^2 + x + 1} $$
$\frac{2x + 2 - 2\sqrt{x}}{x^2 + x + 1}$
More Information
Rationalizing the denominator is a technique used to eliminate radicals from the denominator of a fraction. This makes the expression easier to work with in some contexts.
Tips
A common mistake is to forget to multiply both the numerator and the denominator by the conjugate, thus changing the value of the original expression. Another mistake is incorrectly applying the difference of squares when simplifying the denominator.
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