Problem 68 - Algebra A Combined Function

Question

Find the value of \(x^{2}-3 x+1\) for each given value of \(x\). $$ 5 $$

Step-by-Step Solution

Answer

11

Step 1: Substitute the Value

Insert the given value of \(x\) into the expression. You need to substitute \(x = 5\) into \(x^2 - 3x + 1\).

Step 2: Calculate the Squared Term

Calculate \(5^2\) which gives you 25.

Step 3: Calculate the Linear Term

Calculate \(3 \times 5\) which equals 15.

Step 4: Substitute into Expression

Replace \(x^2\) with 25 and \(3x\) with 15 in the expression: \(25 - 15 + 1\).

Step 5: Simplify the Expression

Perform the arithmetic operation: \(25 - 15 = 10\), then add 1 to obtain 11.

Key Concepts

Substitution Method

The substitution method is a powerful technique often used in algebra to solve or simplify mathematical expressions and equations. This method involves replacing a variable with a specific value.
This can help transform complex expressions into simpler arithmetic calculations. In our example, we are asked to evaluate the quadratic expression \(x^2 - 3x + 1\) for a given value of \(x = 5\).
By substituting \(x\) with \(5\), the original expression becomes \(5^2 - 3 \cdot 5 + 1\).
When performing substitutions:

• Ensure you accurately replace the variable everywhere it appears in the expression.
• Carefully write down each substitution step to avoid mistakes.
• Check your substitution with the initial problem to ensure correctness.

Applying substitution transforms an algebraic expression into one that's easier to solve using basic arithmetic.

Arithmetic Operations

Arithmetic operations refer to basic computations such as addition, subtraction, multiplication, and division. When working with any algebraic expression, particularly after substituting a variable, you will typically use these operations to simplify the expression.
In the exercise, we substitute \(x\) with \(5\) and perform the following arithmetic operations:

• First, calculate \(5^2\), which multiplies \(5\) by itself to yield \(25\).
• Next, compute \(3 \times 5\), resulting in \(15\).
• Finally, substitute these results back into the original expression to simplify it into \(25 - 15 + 1\).

Arithmetic operations are simple yet crucial for breaking down expressions into more manageable pieces.
They allow even complex calculations to be carried out step by manageable step.

Polynomial Evaluation

Evaluating a polynomial means finding the numerical value of the expression when the variables are given specific values. This process is essential for understanding how polynomials behave under various scenarios and can be applied across numerous areas of mathematics.
To evaluate the polynomial \(x^2 - 3x + 1\) for \(x = 5\), you:

• Substitute \(x = 5\) into the polynomial, giving you \(5^2 - 3 \times 5 + 1\).
• Calculate the individual components: \(5^2\) equals \(25\) and \(3 \times 5\) equals \(15\).
• Combine these values to simplify the polynomial to \(25 - 15 + 1\), ultimately resulting in \(11\).

Evaluating polynomials can be straightforward if you methodically follow through the process of substitution and proceed with basic arithmetic operations.
This ensures that you can accurately determine the expression's value for any given input.