f(x) = (3x - 6√x) / (5x^2 - 2)
Understand the Problem
The question presents a function f(x) expressed as a fraction involving polynomial and square root terms. The task is likely to analyze or simplify this function according to algebraic principles.
Answer
The simplified form of the function is $$ f(x) = \frac{3\sqrt{x}(\sqrt{x} - 2)}{5x^2 - 2} $$
Answer for screen readers
The simplified form of the function is $$ f(x) = \frac{3\sqrt{x}(\sqrt{x} - 2)}{5x^2 - 2} $$
Steps to Solve
- Identify the function components
The function is given as ( f(x) = \frac{3x - 6\sqrt{x}}{5x^2 - 2} ). Here, the numerator is ( 3x - 6\sqrt{x} ) and the denominator is ( 5x^2 - 2 ).
- Simplifying the numerator
We can factor the numerator. Notice that both terms in the numerator share a common factor: $$ 3x - 6\sqrt{x} = 3(\sqrt{x})(\sqrt{x} - 2) $$
- Rewriting the function
Now we can rewrite the function using the factored numerator: $$ f(x) = \frac{3\sqrt{x}(\sqrt{x} - 2)}{5x^2 - 2} $$
- Analyzing the denominator
The denominator is ( 5x^2 - 2 ). This part does not seem to factor in a simple way, so we will leave it as is for now.
- Final form of the function
Thus, the simplified form of ( f(x) ) becomes: $$ f(x) = \frac{3\sqrt{x}(\sqrt{x} - 2)}{5x^2 - 2} $$
The simplified form of the function is $$ f(x) = \frac{3\sqrt{x}(\sqrt{x} - 2)}{5x^2 - 2} $$
More Information
The function is now expressed in a simplified form that makes it easier to analyze its properties, such as finding its roots, limits, or potential critical points. The numerator shows that it can equal zero when ( \sqrt{x} - 2 = 0 ) or ( x = 4 ).
Tips
- Forgetting to factor out the common terms in the numerator can lead to a less simplified expression.
- Misapplying the rules for polynomial expressions, especially with roots.
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