Problem 70 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The value of the expression is 19 when \(x = -3\).

Step 1: Substitute the Value of x into the Expression

First, we need to replace the variable \(x\) in the expression \(x^2 - 3x + 1\) with the given value of \(x = -3\). This gives us \((-3)^2 - 3(-3) + 1\).

Step 2: Calculate the Square of x

Calculate \((-3)^2\), which equals 9.

Step 3: Calculate the Product of -3 and x

Calculate \(-3 imes (-3)\), which equals 9. Notice that multiplying two negative numbers results in a positive number.

Step 4: Substitute the Calculated Values Back into the Expression

Substitute the values from Steps 2 and 3 back into the expression: \(9 + 9 + 1\).

Step 5: Perform the Addition

Add the numbers from Step 4 together: \(9 + 9 + 1 = 19\).

Key Concepts

Expression Substitution

Substituting values into expressions is an essential skill in algebra. The idea is simple: whenever you see a variable, replace it with a given number. In our exercise, the variable is \( x \), and the expression is \( x^2 - 3x + 1 \). We're asked to evaluate it when \( x = -3 \). This means every \( x \) in the expression must be replaced by \( -3 \).

Here's how you do it:

• Begin with the original expression: \( x^2 - 3x + 1 \).
• Substitute \( x = -3 \) into the expression, yielding \((-3)^2 - 3(-3) + 1\).

This process converts an algebraic expression into a numerical one, setting the stage for calculation.

Polynomial Evaluation

Understanding polynomial evaluation is crucial when working with algebraic expressions. A polynomial, like \( x^2 - 3x + 1 \), consists of terms with variables raised to varying powers, multiplied by coefficients. Evaluating a polynomial involves computing its value at specific points.

In our case, we evaluate this polynomial by substituting \( x = -3 \).

• The first term \( x^2 \) becomes \((-3)^2\).
• The second term \( -3x \) turns into \(-3(-3)\).
• The constant term \(+1\) simply stays as \+1\.

Now, we have concrete numbers to operate with, transitioning from abstract algebra to clear arithmetic.

Mathematical Operations

Once substitution is complete, the next step involves carrying out basic mathematical operations. These operations simplify the expression step by step.

Let's break down the operations for our example:

• Square the substituted value: Calculate \((-3)^2\), which results in 9.
• Multiply: Calculate \(-3 \times (-3)\), which results in 9. Remember, multiplying two negative numbers gives a positive number.

These operations might seem straightforward, but they reinforce fundamental math skills necessary for more complex algebraic tasks.

Simplification Steps

After performing the calculations, the next step is simplification. This involves combining all the computed values into a single result.

For the expression \((-3)^2 - 3(-3) + 1\), we now have:

• First term: 9
• Second term: 9
• Constant term: 1

Add these results together: \(9 + 9 + 1 = 19\).
Simplification consolidates multiple numbers into a single solution, offering a clear and concise answer to the evaluation problem.