To determine the value of Audrey's stock after two years, we can account for the percentage increase and decrease using an algebraic expression.

Let the initial value of the stock be $x$ dollars.

Step-by-Step Explanation:

1. Increase by 31% in the first year:
   If the stock increased by 31%, then it is worth:
   $x + 0.31x = x(1 + 0.31) = 1.31x$

2. Decrease by 34% in the second year:
   If the stock decreased by 34% in the second year, then it is now worth:
   $1.31x - 0.34(1.31x)$
   Factoring out $1.31x$:
   $1.31x(1 - 0.34) = 1.31x(0.66)$

Final Expression:

The value of the stock after two years is:
$1.31 \cdot 0.66 \cdot x$

Simplifying $1.31 \cdot 0.66$:
$1.31 \cdot 0.66 = 0.8646$

Thus, the value of the stock after two years is:
$0.8646x$

Conclusion:

The expression representing the value of the stock after two years is:
$0.8646x$

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Do you want me to explain why this works with real numbers or clarify any step?

Related Questions:

1. What is the percentage change over the two years combined?
2. If $x = 1000$, what is the stock value after two years?
3. How would the result change if the stock increased by 31% for two consecutive years?
4. What expression would represent a 31% increase followed by a 10% decrease?
5. If Audrey sold the stock after the second year, what percentage of her initial investment did she retain?

Tip:

Percentage increases and decreases are not additive. Always apply each percentage sequentially to the updated value!