Problem 114 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

0

Step 1: Identify the given expression

The expression given is \((x-4)(x-3)\). You will substitute the value of x to evaluate the expression.

Step 2: Substitute the value of x

Given that \(x=4\), substitute 4 into the expression: \((4-4)(4-3)\).

Step 3: Simplify each term inside the parentheses

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

Step 4: Multiply the simplified terms

Now multiply the results: \(0 \times 1 = 0\).

Step 5: Conclusion

The value of the expression \((x-4)(x-3)\) when \(x=4\) is \(0\).

Key Concepts

Substitution in Algebra

Substitution in algebra is a fundamental process that involves replacing a variable with a given value. This makes it easier to simplify or solve expressions.
For example, in the given expression \((x-4)(x-3)\), we are asked to evaluate it when \(x=4\). To do this, we substitute 4 wherever we see \(x\) in the expression.
Substitution simply means:

• Identify the variable in the expression.
• Replace the variable with the given value.
• Proceed to simplify the new, numerical expression.

This process helps to turn a more complex expression into a simpler one, like turning variables into numbers, which are easier to handle.

Simplifying Expressions

Simplifying expressions is a crucial step to making algebraic problems more manageable. Simplification usually involves performing basic arithmetic operations:

• Addition and subtraction
• Multiplication and division

Once you substitute values into the expression, your next step is to simplify the terms inside any parentheses. For example, after substituting \(x=4\) into \((x-4)(x-3)\), we simplify each term inside the parentheses:

• \(4-4=0\)
• \(4-3=1\)

By simplifying these, the expression becomes much simpler, allowing for further operations like multiplication.

Algebraic Multiplication

Multiplication in algebra involves combining simplified terms to find the overall value of an expression. In this particular problem, after substituting \(x=4\) and simplifying the terms inside the parentheses, you get:\((0)(1)\).
The final step involves multiplying these terms together. In this case, multiplication is straightforward:
\(0 \times 1 = 0\).

Here’s a quick reminder when multiplying algebraic terms:

• Always simplify inside parentheses first.
• Then multiply the simplified terms.
• Remember, any number multiplied by zero is zero.

Therefore, the value of the expression \((x-4)(x-3)\) when \(x=4\) is 0.