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Chapter 3: Problem 266

In the following exercises, evaluate each expression. \(3 u^{2}-4 u+5\) when \(u=-3\)

Short Answer

Expert verified
44

Step by step solution

01

Substitute the Value of u

Replace every instance of the variable \(u\) in the expression with the given value \(-3\). The expression becomes: \(3(-3)^{2} - 4(-3) + 5\)
02

Evaluate the Exponent

First, calculate the exponent in the expression: \((-3)^2 = 9\). Now the expression is: \(3 \times 9 - 4(-3) + 5\)
03

Multiply

Perform the multiplication operations: \(3 \times 9 = 27\) \(-4 \times (-3) = 12\). Now the expression is: . \(27 + 12 + 5\)
04

Combine All Terms

Add the results to find the value of the expression: \(27 + 12 + 5 = 44\). . Therefore, the value of \(3u^{2} - 4u + 5\) when \(u = -3\) is \(44\)

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Key Concepts

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

Substitution in Algebra
Substitution in algebra is the process of replacing a variable in an expression with a given value. This is the first critical step when asked to evaluate an expression. The goal is to turn an algebraic expression into a numerical one so that it can be simplified.

For example, consider the expression given in the exercise:
Order of Operations
In mathematics, the order of operations is a set of rules that dictates the sequence in which you should perform calculations within an expression. Following these rules ensures that everyone solves the expression in the same way and gets the same answer.

Consider the expression we worked on earlier: \(3(-3)^2 - 4(-3) + 5\)
To solve this properly, we follow these steps:
  • First, handle any expressions inside parentheses. In our example, \((-3)\) is already simplified.
  • Next, evaluate any exponents. Compute \((-3)^2 = 9\).
  • Then, perform the multiplication and division from left to right. So, we have \(3 \times 9 = 27\) and \(-4 \times (-3) = 12\).
  • Finally, do the addition and subtraction from left to right. So, \(27 + 12 + 5 = 44\).
Remembering these steps will help you simplify any expression accurately.
Simplifying Expressions
Simplifying expressions involves combining like terms and performing basic arithmetic operations such as addition and multiplication. It is the final step after substituting values and respecting the order of operations.

In our example, after substituting \(u = -3\) and following the order of operations, we simplified the terms step by step until we ended up with \(44\). Here's a breakdown:
  • After substitution: \(3(-3)^2 - 4(-3) + 5\)
  • After evaluating the exponent: \(3 \times 9 - 4(-3) + 5\)
  • After multiplication: \(27 + 12 + 5\)
  • Final result after addition: \(44\)
By simplifying expressions in a systematic way, you make sure all terms are correctly combined, and the expression is as simple as possible. Always take your time to check each step carefully.
Following these principles will help you master algebraic expressions in no time!

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