Problem 39 - Elementary Algebra Within Reach

Question

In Exercises 37-44, evaluate the algebraic expression for the given values of the variable(s). \(2 x^{2}-5\) (a) \(x=2\) (b) \(x=3\)

Step-by-Step Solution

Answer

When \(x=2\), the expression equates to \(3\). When \(x=3\), the expression totals to \(13\).

Step 1: Substitute the value \(x=2\)

First, substitute \(x=2\) into the given algebraic expression, \(2 x^{2}-5\), to get \(2(2)^{2}-5\).

Step 2: Calculate the expression for \(x=2\)

Now, calculate the expression. This gives you \(2*4-5\), which simplifies to \(8-5\), and finally to \(3\).

Step 3: Substitute the value \(x=3\)

Replace \(x\) in the original expression with \(3\), leading to \(2 (3)^{2}-5\).

Step 4: Calculate the expression for \(x=3\)

Calculate the expression. This gives you \(2*9-5\), which simplifies to \(18-5\), and finally to \(13\).

Key Concepts

Substitution Method

The substitution method involves replacing a variable within an algebraic expression with a known value. This is a practical way to find the result of an expression when the variables are given certain values. For example, when we are given the expression \(2x^2 - 5\) and the value of \(x\) is 2:

• Replace \(x\) with 2 to rewrite the expression as \(2(2)^2 - 5\).
• You substitute directly into the expression, maintaining the structure of the expression as it was.

This method simplifies the evaluation process by breaking down the problem into more manageable parts.
It is often the first step in solving many algebraic problems, allowing us to move from a general expression to a specific numerical value.

Evaluating Expressions

Evaluating an expression involves calculating its value after substituting all variables with their given numbers. Following substitution, the task is simply to compute the arithmetic. For the expression \(2x^2 - 5\) when \(x = 2\):

• The calculation begins with finding \(2(2)^2\), which results in \(2 \times 4\).
• Then perform the operations in order, resulting in 8.
• Finally, subtract 5 from 8 to get 3.

The same process is repeated if \(x\) is another value, like 3. Each time, the expression is recalculated, demonstrating that the value of the expression changes depending on the variable's value.

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their simplest form. This often makes calculations easier and solutions clearer. For instance, when dealing with \(2x^2 - 5\) and \(x = 3\):

• The expression is simplified to \(2(3)^2 - 5\) first, transforming \(3^2\) into 9.
• Follow through by multiplying, resulting in \(18\).
• Finally, subtracting 5 leaves us with the simplified value, 13.

Simplification not only clarifies each part of a problem but also ensures accurate and efficient processing of the mathematics involved. This is crucial when working with complex expressions or larger sets of data.