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Chapter 3: Problem 265

In the following exercises, evaluate each expression. \(2 w^{2}-3 w+7\) when \(w=-2\)

Short Answer

Expert verified
21

Step by step solution

01

Substitute the Value of w

First, substitute the given value of w, which is -2, into the expression. The expression is: \(2 w^{2} - 3 w + 7\). After substitution, it becomes \(2 (-2)^{2} - 3 (-2) + 7\).
02

Evaluate the Squared Term

Next, we need to evaluate the squared term. Calculate \((-2)^{2}\), which equals 4. Therefore, the expression now is \(2 \times 4 - 3 (-2) + 7\).
03

Multiply the Coefficient with the Squared Value

Now, multiply the coefficient 2 by the squared value 4. This gives us 8. So the expression is now \(8 - 3 (-2) + 7\).
04

Evaluate the Negative Multiplication

Next, evaluate the product of -3 and -2. Multiplying these values gives us 6. Thus the expression is now \(8 + 6 + 7\).
05

Add the Values Together

Finally, add the values together: 8 + 6 + 7. This equals 21.

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Key Concepts

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

substitution
Substitution is the process of replacing a variable in an algebraic expression with a given value. In our example, we substituted w with -2. So, the original expression \(2 w^{2} - 3 w + 7\) changes to \(2 (-2)^{2} - 3 (-2) + 7\).
Substitution helps simplify expressions by eliminating variables,
making it easier to perform the remaining steps.
order of operations
To ensure accurate results when evaluating expressions, always follow the order of operations. The common acronym to remember is PEMDAS:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
In our example, after substitution, the expression is \(2 (-2)^{2} - 3 (-2) + 7\).
We first evaluate the exponent \((-2)^{2}\), followed by multiplication and then addition/subtraction.
negative numbers multiplication
When multiplying negative numbers, remember the rules:
  • Negative 脳 Negative = Positive
  • Negative 脳 Positive = Negative
In our example, we needed to multiply \(-3\) and \(-2\). According to the rule, two negatives make a positive, so \(-3 脳 -2 = 6\).
This critical step transforms the expression to \(8 + 6 + 7\).
addition of integers
Adding integers involves combining positive and negative numbers. The final step is to add the calculated values together:
  • 8 + 6 = 14
  • 14 + 7 = 21
It's important to perform addition carefully to get the correct result.
Hence, our final expression evaluates to 21, demonstrating the importance of meticulous addition.

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Most popular questions from this chapter

In the following exercises, evaluate each expression. $$ -15-3 u^{2} \text { when } u=-5 $$

In the following exercises, evaluate each expression. $$ \begin{array}{l}{10-y \text { when }} \\ {\text { (a) } y=15} \\ {\text { (b) } y=-16}\end{array} $$

In the following exercises, evaluate. $$ 3 m-2 n \text { when } m=6, n=-8 $$

In the following exercises, determine whether each number is a solution of the given equation. $$ 8 u+24=-32 $$ $$ \begin{array}{l}{\text { (a) } u=-7} \\ {\text { (b) } u=-1 \quad \text { (c) } u=7}\end{array} $$

In the following exercises, simplify. $$ -7-(-3) $$
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