Problem 69 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The value is 5.

Step 1: Substitute the Given Value of x

We are given that \(x = -1\). Substitute \(-1\) into the expression \(x^2 - 3x + 1\). This gives us \((-1)^2 - 3(-1) + 1\).

Step 2: Calculate the Square of x

Calculate \((-1)^2\). Since \(-1\) squared is \(1\), we have: \(1 - 3(-1) + 1\).

Step 3: Multiply x by -3

Calculate \(-3(-1)\). Multiplying \(-3\) by \(-1\) gives \(+3\), so the expression becomes \(1 + 3 + 1\).

Step 4: Add the Terms Together

Now, add \(1 + 3 + 1\) to find the final value. \(1 + 3 + 1 = 5\). Hence, the value of the expression \(x^2 - 3x + 1\) for \(x = -1\) is \(5\).

Key Concepts

Substitution Method

The substitution method is a technique used to evaluate algebraic expressions by replacing variables with specific values. This method is highly useful in both algebra and calculus for simplifying calculations.Key steps involve:

• Identifying the variable and its given value in the expression.
• Substitute the given value into the expression wherever the variable appears.
• Perform arithmetic operations to simplify and find the result.

In our original exercise, we used the substitution method to find the value of the expression \(x^2 - 3x + 1\) when \(x = -1\). By substituting \(-1\) in place of \(x\), the expression becomes \((-1)^2 - 3(-1) + 1\). This transformed expression is then evaluated to determine the value.

Simplifying Expressions

Simplifying expressions involves performing arithmetic operations to rewrite an algebraic expression in its simplest form. This process makes it easier to work with the expression and reach solutions more efficiently.Important principles include:

• Handling each mathematical operation step by step.
• Combining like terms and reducing the expression to minimal terms.
• Following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

In our problem, after substitution, the expression \((-1)^2 - 3(-1) + 1\) needs to be simplified. We carried out the calculations as follows:- Firstly, \((-1)^2\) is calculated, resulting in \(1\).- Next, \(-3(-1)\) is worked out, giving \(+3\).- Finally, all terms \(1\), \(+3\), and \(1\) are added together resulting in \(5\). This is the simplest form of the expression for \(x = -1\).

Negative Numbers in Algebra

Working with negative numbers is a fundamental part of algebra. Negative numbers can initially pose challenges, but understanding how they interact in equations and expressions helps simplify complex problems.Key points to remember:

• Squaring a negative number, such as \((-1)^2\), results in a positive number.
• Multiplying two negative numbers, like \(-3\) and \(-1\), results in a positive number.
• Be cautious of signs during operations; misplacing a negative sign can change the outcome entirely.

In the provided example, we demonstrated these principles:- When \((-1)^2\) was calculated, the result was \(1\).- The operation \(-3(-1)\) produced \(+3\) since multiplying two negatives gives a positive.This understanding allowed us to correctly evaluate the expression and reach the correct outcome. Managing negative numbers requires careful attention to ensure accuracy in algebraic calculations.