Problem 78 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each algebraic expression for the given value of the variable. $$-x^{2}-14 x ; x=-1$$

Step-by-Step Solution

Answer

The evaluated expression for \(x=-1\) in \(-x^{2}-14x\) is \(13\).

Step 1: Substitute the Value

Replace \(x\) in the algebraic expression \(-x^{2}-14x\) with the given value. This results in \(-(-1)^{2}-14*(-1)\).

Step 2: Evaluate the Powers

Both the \(-1\) in the first term is squared. This leads to the expression \(-1^{2}-14*(-1)\), which simplifies to \(-1-14*(-1)\).

Step 3: Perform Multiplications

Perform multiplication in the second term. This results in \(-1 + 14\).

Step 4: Perform the Addition

Lastly, add the two terms \(-1 + 14\) together to get the final result, which is \(13\).

Key Concepts

Variable Substitution

One of the first steps when evaluating an algebraic expression is variable substitution. This means you replace the variable in an expression with a given numerical value. When you see a problem like -\(x^2 - 14x\), and you're told \(x = -1\), substitute \(-1\) for every \(x\) in the expression.This results in replacing each occurrence of \(x\) with \(-1\), leading to -(-1)^2 - 14(-1).

• This makes the tricky parts clearer, breaking complex expressions into numerical calculations.
• Whenever variable values are provided, substitution is a straightforward way to start solving the expression.

Proper substitution ensures that you're working with numbers only, simplifying further operations in calculation.

Exponents

Exponents signify how many times to multiply a number by itself. In expressions like -(-1)^2 - 14(-1), exponents play a crucial role. Here, (-1)^2 means multiply -1 by itself: -1 × -1 = 1.

• Remember, when raising negative numbers to even exponents, the result is a positive number.
• Conversely, raising negative numbers to odd exponents results in a negative number.

Pay attention to exponents in expressions, because they change the numbers drastically, altering the whole equation as a result.

Multiplication

In our journey of solving algebraic expressions, multiplication often links different terms together. Once variable substitution and exponent calculations are out of the way, next comes multiplication. In the expression from our example: -(-1)^2 - 14(-1), proved straightforward. We multiply 14 times -1, which equals -14.

• Remember the basic rule: Positive times a negative, or negative times positive always yields a negative product.
• Two negatives multiply to give a positive result.

Master these multiplication basics to progress smoothly through solving expressions.

Addition

After managing all other operations in an algebraic expression, addition helps combine our terms for the final answer. In the previous steps of our example, we reached: -1 + 14. Performing this last operation is simple:

• Adding -1 and 14 together.
• Because one number is negative and the other positive, this is equivalent to subtraction: 14 - 1, resulting in 13.

In such expressions, the order of addition might involve negatives mid-way, but by treating it as combining values on number lines, you arrive at accurate results.