Problem 113 - Elementary Algebra

Question

An expression is \((x-4)(x-3)\). Evaluate this expression when \(x=3\).

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Substitute the value of x

Replace the variable x with the given value, which is 3, in the expression \((x-4)(x-3)\).

Step 2: Simplify each term

Simplify the expressions inside the parentheses. Calculate \((3-4)\) and \((3-3)\).

Step 3: Calculate the simplified terms

Evaluate the simplified terms: \((3-4) = -1\) and \((3-3) = 0\).

Step 4: Multiply the results

Multiply the evaluated terms together: \(-1 \times 0 = 0\).

Key Concepts

Substitution of Variables

To evaluate an algebraic expression, the first step is to perform the substitution of variables. This means replacing the variable in the expression with the value provided.
In the given exercise, we have the expression \((x-4)(x-3)\), and we must evaluate this when \(x=3\).
So, we substitute 3 for every occurrence of \(x\) in the expression. This results in \( (3-4)(3-3) \).
Once the substitution is complete, we proceed to the next steps. Substitution forms the foundation of solving any algebraic expression because it transforms a general expression into a simpler numeric form.

Simplifying Expressions

After the substitution, simplifying expressions is the next step. Simplification makes the expression easier to work with by breaking it down into simpler parts you can easily calculate.
Take the expression we derived from substituting \(x=3\). This gives \( (3-4)(3-3) \).
Inside the parentheses, we have two operations: \(3-4\) and \(3-3\).
We now simplify each term:

• \(3-4 = -1\)
• \(3-3 = 0\)

When you simplify these terms, it breaks the expression down into manageable pieces. This step is crucial because it prepares the expression for further operations like multiplication.

Multiplication of Integers

Finally, after simplifying the expression, we handle the multiplication of integers. This is the last step in our original problem.
From our earlier result of simplifying the expressions, we have \(-1\) and \(0\).
So, we need to multiply these simplified terms: \(-1 \times 0 = 0\).
The multiplication of integers can alter the sign and size of numbers, but anything multiplied by zero is zero. Hence, the final result of our expression is zero.
The multiplication step essentially combines the simplified parts into a final solution. Always double-check your simplified terms before this step to ensure accuracy.