Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression for each value of \(x\). (If not possible, state the reason.)\(\begin{array}{lll}x^{2}-3 x+4 & \text { (a) } x=-2 & \text { (b) } x=2\end{array}\)

Step-by-Step Solution

Answer

When \(x = -2\), the expression evaluates to \(14\). When \(x = 2\), the expression evaluates to \(2\).

Step 1: Substitute -2 for x

First, substitute \(-2\) for \(x\) in our equation \((x^{2} -3x + 4\). So it becomes: \((-2)^{2}-3(-2)+4\)

Step 2: Evaluate Expression for x = -2

Now evaluate the expression. \((-2)^2 = 4\), \(-3*-2 = 6\), so the expression becomes \(4 + 6 + 4\). Adding this all together gives the result as \(14\).

Step 3: Substitute 2 for x

Next, substitute \(2\) for \(x\) into the equation, so it becomes \((2)^2 -3*2 + 4\)

Step 4: Evaluate expression for x = 2

Evaluating the expression, we find \((2)^2 = 4\), \(-3*2 = -6\), so our expression becomes \(4 - 6 + 4\). Adding this together gives us as \(2\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra where we replace a variable, often denoted as \( x \), with a specific value to simplify or solve an algebraic expression or equation. Let’s break it down into a few simple steps:

• Identify the given expression or equation, such as \( x^2 - 3x + 4 \).
• Substitute the given value(s) for \( x \) into the expression. For example, substitute \( -2 \) or \( 2 \) for \( x \).
• Recalculate the expression to find a numeric result.

This method is extremely useful for evaluating expressions at specific values, allowing you to solve problems more efficiently. Substitution helps you convert complex algebraic expressions into simple arithmetic calculations. By focusing on one variable at a time, you are able to better manage complex interactions within an expression. Overall, mastering this technique will enhance your problem-solving skills in algebra.

Polynomial Evaluation

Polynomial evaluation involves computing the value of a polynomial for a particular value of its variable. A polynomial is an expression consisting of variables and coefficients, like \( x^2 - 3x + 4 \), which you'll often encounter in middle and high school math courses.

• Structure of a Polynomial: It consists of terms that are added or subtracted. Each term includes a variable raised to an exponent and a coefficient. For instance, in \( x^2 - 3x + 4 \), \( x^2 \) and \(-3x\) are terms with coefficients 1 and -3, respectively.
• Evaluation: To evaluate a polynomial, simply substitute the variable with a specific value and carry out the arithmetic operations.

This process applies the substitution method within a context of a polynomial, making it a simple yet powerful tool for determining the polynomial's value at selected points. Mastery of this will build a strong foundation for solving equations and understanding functions.

Quadratic Expression

A quadratic expression like \( x^2 - 3x + 4 \) is a type of polynomial with a degree of 2, meaning the highest power of the variable \( x \) is 2. Quadratic expressions are central to many aspects of algebra due to their wide range of applications:

• Structure: A typical quadratic expression is written as \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants. In this form, \( a = 1 \), \( b = -3 \), and \( c = 4 \).
• Forms of a Quadratic: These can often be factored, or rewritten in different forms to make solving or graphing easier. However, not all quadratics will factor neatly.
• Relevance: Quadratics appear in various mathematical problems, including physics and engineering scenarios. Solving these helps in finding roots or zeroes, which are the values of \( x \) that make the expression equal to zero.

Understanding quadratic expressions will open the door to more advanced concepts in mathematics and improve your analytical skills.