Problem 67 - Algebra A Combined Function

Question

Find the value of \(x^{2}-3 x+1\) for each given value of \(x\). $$ 2 $$

Step-by-Step Solution

Answer

The value is \(-1\).

Step 1 - Understand the Expression

The given expression is a quadratic expression, \( x^2 - 3x + 1 \). Our task is to evaluate the expression for specific values of \( x \).

Step 2 - Substitute the Given Value

We need to substitute the given value of \( x = 2 \) into the expression. The expression becomes: \( (2)^2 - 3(2) + 1 \).

Step 3 - Evaluate the Squared Term

Calculate \( 2^2 \): This equals \( 4 \). The expression is now: \( 4 - 3(2) + 1 \).

Step 4 - Evaluate the Multiplication Term

Calculate \( 3 \times 2 \): This equals \( 6 \). Substitute back, so the expression is now \( 4 - 6 + 1 \).

Step 5 - Perform the Subtraction and Addition

First, perform the subtraction: \( 4 - 6 = -2 \). Then add \( 1 \) to \(-2\): \(-2 + 1 = -1 \).

Step 6 - State the Final Result

The evaluated result of the expression \( x^2 - 3x + 1 \) when \( x = 2 \) is \(-1\).

Key Concepts

Substitution Method

The substitution method is a common technique used in algebra to simplify and solve equations or expressions. In the context of quadratic expressions like \( x^2 - 3x + 1 \), substitution helps us find the specific value of an expression by replacing the variable, \(x\), with a given number.

This method involves a straightforward approach. Begin by taking the given expression and the specific value assigned to the variable. Here, \( x = 2 \). You then "substitute" this value into every instance of \(x\) in the expression.

• Identify the expression: \( x^2 - 3x + 1 \)
• Substitute \( x = 2 \) into the expression: \( (2)^2 - 3(2) + 1 \)

This substitution transforms the expression into numerical terms that you can easily evaluate. By doing this, the variable \(x\) vanishes, and you're left with arithmetic to perform.

Evaluation of Expressions

Evaluating expressions is a fundamental skill in algebra, and it involves performing the arithmetic operations present in the expression once the substitution method has been used. The goal is to simplify the expression to find a single numerical result.

After substituting \( x = 2 \) into our expression, we have \( (2)^2 - 3(2) + 1 \). To evaluate this, tackle each operation step by step:

• Square operation: calculate \( (2)^2 \), which results in \( 4 \).
• Multiplication: calculate \( 3 \times 2 \), which results in \( 6 \).
• Combine the results using subtraction and addition: \( 4 - 6 + 1 \).

Following these steps helps ensure that each part of the expression is correctly calculated and simplifies the process of arriving at the final result, which in this example is \(-1\).

Algebraic Operations

Algebraic operations refer to the basic mathematical procedures such as addition, subtraction, multiplication, and squaring that are used to manipulate algebraic expressions. Understanding these is crucial when working with any type of equation, especially quadratic expressions.

For the expression \( x^2 - 3x + 1 \), several operations are performed:

• Squaring: raising 2 to the power of 2 to manage \( x^2 \).
• Multiplication: \( 3 \times 2 \) for the linear term \(3x\).
• Subtraction and addition: \( 4 - 6 + 1 \) to consolidate terms.

In each step, follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures the calculation is organized and accurate, leading to the correct evaluation of expressions like the provided quadratic one.