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Chapter 3: Problem 264

In the following exercises, evaluate each expression. \(q^{2}-2 q+9\) when \(q=-2\)

Short Answer

Expert verified
The value of the expression is 17.

Step by step solution

01

- Substitute the value of q

Substitute the given value of \(q = -2\) into the expression \(q^{2}-2q+9\).
02

- Evaluate \(q^2\)

Calculate \((-2)^{2}\). The result is: \((-2)^{2} = 4\).
03

- Evaluate \(-2q\)

Calculate \(-2q\). Substitute \(q = -2\): \(-2(-2) = 4\).
04

- Combine the results

Combine the results from Steps 2 and 3, and add 9: \(4 + 4 + 9\).
05

- Simplify the expression

Add the values together: \(4 + 4 + 9 = 17\).

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Key Concepts

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

Substitution Method
The substitution method in algebra involves replacing a variable with a given number. This method helps simplify and solve expressions or equations. In our example, we started with the expression \(q^{2}-2q+9\) and the value \(q = -2\). By substituting \(q = -2\) into the expression, we rewrite \(q^{2}\) as \((-2)^{2}\), \(-2q\) as \(-2 \cdot -2\), and 9 remains as is.

This substitution allows us to replace abstract variables with concrete numbers, making calculations straightforward.

The steps are:
  • Identify the variable in the expression
  • Replace the variable with the given value
  • Simplify the expression
By doing this, we transformed an algebraic expression into a simple arithmetic problem.
Simplifying Algebraic Expressions
To simplify algebraic expressions means to reduce them to their simplest form. This includes performing operations like addition, subtraction, multiplication, or division.

In our exercise, we first calculated \((-2)^{2}\) to get 4. Next, we found the value of \(-2 \cdot -2\), which is also 4. Finally, we added these results to get \(4 + 4 + 9\).

Simplifying involves combining like terms and performing any operations as needed. Here are some general steps to follow:
  • Substitute known values into the expression
  • Evaluate powers and perform multiplications or divisions first (following the order of operations)
  • Combine like terms by adding or subtracting them
This ensures our expression is as straightforward as possible and easy to understand.
Basic Algebra
Basic algebra involves understanding and using algebraic expressions, equations, and functions. It's the foundation of more complex mathematics.

In basic algebra, we frequently encounter expressions with variables that we need to evaluate or simplify.

Let's break down the key concepts:
  • Variables: Symbols (like \(q\) or \(x\)) representing unknown values.
  • Expressions: Combinations of variables, numbers, and operations (e.g., \(q^{2}-2q+9\)).
  • Order of Operations: Rules to follow for simplifying expressions (PEMDAS/BODMAS).
  • Substitution: Replacing variables with specific values to evaluate expressions.

Practicing these basics makes it easier to handle more advanced topics in algebra and other mathematical disciplines. Understanding how to evaluate and simplify expressions is crucial for solving real-world problems.

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Most popular questions from this chapter

In the following exercises, solve the given applications. $$ -1 \cdot 6 $$

In the following exercises, simplify. $$ -8+6 $$

In the following exercises, compare the numbers, using \(<\) or \(>\) or=. (a) \(-5\\_|-5|\) (b) \(-|-2|\\_-2\)

In the following exercises, evaluate each expression. $$ 5 a+8 b \text { when } a=-2, b=-6 $$

In the following exercises, determine whether each number is a solution of the given equation. $$ 5 x-10=-35 $$ $$ \begin{array}{l}{\text { (a) } x=-9} \\ {\text { (b) } x=-5 \quad \text { (c) } x=5}\end{array} $$
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