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Chapter 3: Problem 47

Explain how to evaluate an expression. $$a^{2}+9 a-2 ; a=-2$$

Short Answer

Expert verified
Question: Evaluate the expression $$a^{2} + 9a - 2$$ when $$a = -2$$. Answer: The value of the expression is $$-16$$.

Step by step solution

01

Substitute the value of a

Replace the value of a in the given expression with -2. So, the expression becomes: \((-2)^{2} + 9(-2) - 2\)
02

Simplify the expression

Evaluate the expression by following the order of operations which are Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS). 1. Evaluate the exponent: \(4 + 9(-2) - 2\) 2. Proceed with multiplication: \(4 - 18 - 2\) 3. Finally, perform addition and subtraction: \((4 - 18) - 2\) \((-14) - 2\)
03

Write the final value

After performing all the operations, we get the final value as: \(-16\) So, the value of the expression $$a^{2} + 9a - 2$$ when $$a = -2$$ is $$-16$$.

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Key Concepts

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

Substitution in Algebra
Substitution is a fundamental concept in algebra that involves replacing a variable with a specific value. In our example, the expression is \(a^2 + 9a - 2\) where \(a = -2\).
By substituting, we replace each occurrence of \(a\) with \(-2\).
This transforms the expression into \((-2)^2 + 9(-2) - 2\).
This straightforward substitution allows us to work directly with numbers rather than variables, making it easier to simplify the expression.
Remember, proper substitution helps in seamlessly carrying out further algebraic operations.
Order of Operations
The order of operations is crucial when evaluating algebraic expressions to ensure accurate results.
This rule is commonly remembered by the acronym PEMDAS:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
In the given expression \((-2)^2 + 9(-2) - 2\), we first deal with the exponent \((-2)^2\) which equals \(4\).
Next, we perform the multiplication \(9 \times (-2) = -18\).
Finally, we handle addition and subtraction, calculating \(4 - 18 - 2\).
By following the correct order, we avoid common errors and ensure that the expression is simplified accurately.
Simplifying Algebraic Expressions
Simplifying expressions means reducing them to their simplest form to make them easier to work with.
In this process, after following the order of operations, the expression \(4 - 18 - 2\) can be further simplified.
Start by solving within the parentheses: \((4 - 18)\) which results in \(-14\).
Next, subtract \(2\) from \(-14\) to get \(-16\).
Simplification helps to make expressions less complex, helping us arrive at the final solution: \(-16\).
Knowing how to simplify expressions is essential for handling complex problems easily and accurately.

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