Problem 104 - Elementary Algebra Within Reach

Question

In Exercises 103 and 104, evaluate the algebraic expression for the given values of the variable. \(2 x^{2}+3\) (a) \(x=3\) (b) \(x=-4\)

Step-by-Step Solution

Answer

The value of the expression \(2 x^{2}+3\) is 21 when \(x=3\) and 35 when \(x=-4\).

Step 1: Substitution and Evaluation for \(x=3\)

Substitute \(x = 3\) into the algebraic expression, giving: \(2*(3)^{2}+3 = 2*9+3 = 18+3 = 21\)

Step 2: Substitution and Evaluation for \(x=-4\)

Substitute \(x = -4\) into the algebraic expression, giving: \(2*(-4)^{2}+3 = 2*16+3 = 32+3 = 35\)

Key Concepts

Substitution in Algebra

Substitution in algebra involves replacing a variable with a given number. In the exercise above, we are asked to evaluate the expression \(2x^{2}+3\) for different values of \(x\). This technique is essential, as it transforms an abstract expression into a tangible number that can be easily manipulated.

The process is rather straightforward:

• Identify the variable in your algebraic expression.
• Replace this variable with the specified value.
• Recalculate the expression using basic arithmetic operations.

For instance, when \(x=3\), substitute \(3\) into the expression to get \(2(3)^{2} + 3\). Similarly, perform the same with \(x=-4\) to get the second result.

Substitution is a crucial skill in algebra, helping deal with more complex equations by breaking them into manageable parts.

Algebraic Evaluation

Algebraic evaluation is the process of calculating the value of an algebraic expression once the variables have been substituted. This is essentially performing arithmetic operations on numbers after replacing the variables.

Let’s go through this with our example expression \(2x^{2}+3\). After substitution, you perform the operations following the order of:

• Exponents (calculate \(x^{2}\))
• Multiplication/Division (calculate \(2 \times x^{2}\))
• Addition/Subtraction (add the constant, in this case, 3)

For instance, after substituting \(x=3\), you get:

• Step 1: Calculate \(3^{2} = 9\)
• Step 2: Multiply by 2 to get \(2 \times 9 = 18\)
• Step 3: Add 3 to conclude with \(18 + 3 = 21\)

Following the same steps with \(x=-4\), you achieve a similar evaluation.

Understanding algebraic evaluation ensures clarity when handling expressions involving any number of mathematical operations.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and mathematical operations. They are a fundamental component in algebra and many other areas of mathematics.

Mathematical expressions, such as \(2x^{2}+3\), provide a concise way of representing mathematical ideas, where variables can be modified for various scenarios. In this expression:

• \(2x^{2}\) signifies that the variable \(x\) is squared and then multiplied by 2.
• The \(+3\) is a constant added to the result of \(2x^{2}\).

Expressions are not equations— they don't state equality with another quantity. Instead, they are standalone components that can be evaluated once variables are assigned specific values.

Knowing how to work with mathematical expressions opens up opportunities to solve a wide range of problems, from simple arithmetic to advanced mathematics.