Problem 151 - Introductory Algebra for College Students

Question

Exercises 150–152 will help you prepare for the material covered in the next section. Evaluate \(2 x^{2}+7 x-4\) for \(x=\frac{1}{2}\)

Step-by-Step Solution

Answer

The evaluated result of the expression \(2x^{2}+7x-4\) for \(x = \frac{1}{2}\) is \(\frac{1}{2}\).

Step 1: Substitution

First, substitute \(\frac{1}{2}\) into the equation in place of \(x\), so: \(2(\frac{1}{2})^{2}+7(\frac{1}{2})-4\).

Step 2: Simplify \(\frac{1}{2}^{2}\) and multiply

Simplify the exponent making \(\frac{1}{2}^{2} = \frac{1}{4}\), then multiply it by 2, that gives us: \(2 \cdot \frac{1}{4} = \frac{1}{2}\). Now, the equation should look like: \(\frac{1}{2} + 7(\frac{1}{2}) - 4\)

Step 3: Simplify \(7\cdot\frac{1}{2}\)

Multiply 7 with \(\frac{1}{2}\), which equals \(\frac{7}{2}\) or 3.5. Update the equation to: \(\frac{1}{2} + \frac{7}{2} - 4\)

Step 4: Conversion and Subtraction

Convert 4 into a fraction with common denominator of 2 which is \(\frac{8}{2}\) and Subtract \(\frac{8}{2}\) from the sum of \(\frac{1}{2} + \frac{7}{2}\). This gives us \(\frac{8}{2} - 4 = \frac{1}{2}\)

Key Concepts

Substitution Method

The substitution method is a crucial algebraic technique used to evaluate expressions, especially polynomials. It involves replacing variables in an algebraic expression with their numerical counterparts.
For example, if you're asked to evaluate the polynomial expression when a specific value, such as \( x= \frac{1}{2} \) is substituted for \( x \), you simply replace all instances of \( x \) with \( \frac{1}{2} \). This is the first step you would take to see the effect of that particular value on the overall expression. By methodically replacing variables with numbers, you're turning an abstract algebraic concept into a concrete numerical problem that can be solved using basic arithmetic.

The practicality of the substitution method lies in its ability to simplify complex polynomial problems into manageable calculations, making them easier to tackle.

Simplifying Exponents

Square of a Fraction

Understanding how to simplify exponents is essential when you're working with polynomials. For example, raising a fraction to a power, as in \( \left(\frac{1}{2}\right)^2 \), means that you'll multiply the fraction by itself. Here, the square of \( \frac{1}{2} \) results in \( \frac{1}{4} \).
It's important to take this step slowly and ensure the accuracy of your simplification since exponents can drastically change a value. Always remember that simplifying exponents is about following the rules of indices to get to a reduced, more manageable form that will integrate smoothly into the polynomial you are evaluating.

Polynomial Functions

Polynomial functions consist of variables raised to whole number exponents and constants combined using addition, subtraction, multiplication, or division, but not division by a variable.
For example, in the polynomial \( 2x^2 + 7x - 4 \), we have three terms: each has its own coefficient and variable with exponent. These polynomials are the building blocks of algebra and play a pivotal role in mathematical modeling.

When evaluating these functions, it's imperative to consider the role of each term and how it contributes to the value of the polynomial for a given \( x \). It's a step-by-step process—first, substituting values for the variable (using the substitution method), then simplifying terms separately, and finally consolidating them to find the polynomial’s result.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations that represent a particular value for given variable inputs.
For instance, \( 2x^2 + 7x - 4 \) is an algebraic expression where \( x \) can assume any value, transforming the abstract into a specific quantity once evaluated. The true power of algebraic expressions lies in their generality—they can model relationships across a spectrum of real-world applications, from physics to finance.

Evaluating algebraic expressions allows us to interpret the relationships between variables and constants. Much like a recipe, understanding each ingredient (constant, variable, and operation) helps us to compute the final result successfully.