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Chapter 3: Problem 80

Find the value of \(x^{2}-3 x+1\) for each given value of \(x .\) See Section 1.7. $$ 5 $$

Short Answer

Expert verified
The value is 11 when \(x = 5\).

Step by step solution

01

Understanding the Problem

We need to find the value of the expression \(x^2 - 3x + 1\) when \(x = 5\). This means we substitute the given value of \(x\) into the expression and simplify.
02

Substitute the Value

Substitute \(x = 5\) into the expression. Replace every instance of \(x\) in \(x^2 - 3x + 1\) with \(5\). This gives us the expression \(5^2 - 3(5) + 1\).
03

Calculate the Square

Calculate \(5^2\). This means multiplying 5 by itself: \(5 \times 5 = 25\).
04

Calculate the Product

Calculate \(-3 \times 5\). This requires multiplying -3 by 5, which results in \(-15\).
05

Addition and Subtraction

Now, evaluate the expression \(25 - 15 + 1\). First, subtract 15 from 25 to get 10. Then, add 1 to 10 to get 11.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution
Substitution is a fundamental concept in algebra that involves replacing variables with given values. In this exercise, we're working with the quadratic expression \(x^2 - 3x + 1\). The goal is to find its value when \(x\) is known. Here, we substitute \(x = 5\) into the expression. This process involves replacing every occurrence of \(x\) with \(5\). The expression becomes \(5^2 - 3(5) + 1\).
  • Substitution helps in simplifying expressions to numbers.
  • It allows us to evaluate expressions for specific values.
By substituting, we transform an algebraic expression into an arithmetic one that can be easily calculated. This is crucial in solving equations and evaluating expressions.
Simplification
Simplification involves reducing an expression into its simplest form. After substitution, the next step in our exercise is simplification. We've replaced \(x\) with \(5\) to get \(5^2 - 3(5) + 1\). Now, we need to simplify this expression by performing arithmetic operations.
  • Calculate \(5^2\), which is \(25\).
  • Calculate \(-3 \times 5\), giving us \(-15\).
After these calculations, the expression transforms into \(25 - 15 + 1\). Simplification helps break down complex expressions by carrying out each operation step by step. It's an essential skill to develop for handling more intricate algebraic problems.
Evaluating Expressions
Evaluating expressions means computing their value after substitution and simplification. In our example, we have simplified the expression to \(25 - 15 + 1\). Now, we need to evaluate this expression by performing the arithmetic operations. First, subtract \(15\) from \(25\) to get \(10\).
  • Subtraction: \(25 - 15 = 10\).
  • Addition: \(10 + 1 = 11\).
The final evaluated value is \(11\). Evaluating expressions is the process of finding a numerical value and involves executing operations like addition and subtraction until the expression reduces to a single value. It's a crucial final step to determining the outcome of an algebraic expression.

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