Simplify the expression: \frac{2}{(x+1)+\sqrt{x}}
Understand the Problem
The question requires simplifying or rationalizing the denominator of the given expression. We will focus on manipulating the expression to remove the square root from the denominator.
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 both numerator and denominator by the conjugate
Multiply the given expression by $\frac{(x+1) - \sqrt{x}}{(x+1) - \sqrt{x}}$: $$ \frac{2}{(x+1)+\sqrt{x}} \cdot \frac{(x+1)-\sqrt{x}}{(x+1)-\sqrt{x}} $$
- Expand the numerator
Multiply the numerator: $$ 2 \cdot [(x+1) - \sqrt{x}] = 2(x+1) - 2\sqrt{x} $$
- Expand the denominator
Multiply the denominator using the difference of squares formula, $(a+b)(a-b) = a^2 - b^2$. Let $a = (x+1)$ and $b = \sqrt{x}$. $$ [(x+1) + \sqrt{x}] \cdot [(x+1) - \sqrt{x}] = (x+1)^2 - (\sqrt{x})^2 $$
- Simplify the denominator
Expand and simplify the denominator: $$ (x+1)^2 - (\sqrt{x})^2 = (x^2 + 2x + 1) - x = x^2 + x + 1 $$
- Write out the rationalized expression
Combine the simplified numerator and denominator to get the rationalized expression: $$ \frac{2(x+1) - 2\sqrt{x}}{x^2 + x + 1} $$
- Final Answer
The final answer 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 means eliminating any radical expressions (like square roots or cube roots) from the denominator of a fraction. This is often done to simplify expressions or to make them easier to work with, especially when performing further calculations or comparisons.
Tips
A common mistake is not correctly identifying the conjugate. Another common mistake is incorrectly applying the difference of squares formula when multiplying the denominator and its conjugate. Also, students can forget to distribute the '2' in the numerator correctly after multiplying by the conjugate.
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