An algebraic expression is a mathematical phrase that can include numbers, variables (like $x$ or $y$), and operations (like addition, subtraction, multiplication, and division). Unlike equations, algebraic expressions do not have an equal sign and are not meant to be solved for an exact value; instead, they are simplified or evaluated for specific variable values.

Here's an example of an algebraic expression:

$3x^2 - 5x + 2$

Let's go through a step-by-step process of simplifying and evaluating this expression.

Step 1: Simplifying the Expression

First, consider whether the expression can be simplified further. Simplifying means combining like terms (terms with the same variable raised to the same power). In this case, the expression is already simplified because there are no like terms to combine:

$3x^2 - 5x + 2$

Step 2: Evaluating the Expression for a Specific Value of $x$

Let's evaluate the expression for a specific value of $x$. Suppose $x = 2$. To evaluate the expression, substitute $x = 2$ into the expression:

$3(2)^2 - 5(2) + 2$

Step 3: Performing the Calculations

Now, perform the calculations step by step:

1. Calculate the exponent: $2^2 = 4$

2. Multiply the coefficient by the calculated exponent: $3 \times 4 = 12$

3. Multiply the coefficient by the variable: $5 \times 2 = 10$

4. Substitute these results back into the expression: $12 - 10 + 2$

Step 4: Simplify the Expression

Finally, simplify the expression by performing the addition and subtraction:

$12 - 10 = 2$ $2 + 2 = 4$

Final Answer

The value of the expression $3x^2 - 5x + 2$ when $x = 2$ is 4.

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Would you like more details on any part of the process or have any questions?

Here are some related questions you might find interesting:

1. How do you factorize a quadratic expression like $3x^2 - 5x + 2$?
2. What is the difference between an algebraic expression and an algebraic equation?
3. How can you solve for $x$ if the expression is set equal to a number (e.g., $3x^2 - 5x + 2 = 0$)?
4. How does changing the value of $x$ affect the evaluation of an algebraic expression?
5. What is the significance of the coefficients in an algebraic expression?
6. Can an algebraic expression include division and fractional terms?
7. How would the expression change if you introduced another variable, like $y$?
8. What is the process to simplify a more complex algebraic expression?

Tip: When evaluating algebraic expressions, always perform the operations in the correct order (exponents, multiplication/division, addition/subtraction) to avoid errors.