Problem 46 - Prealgebra and Introductory Algebra

Question

Evaluate the variable expression for \(a=-2, b=4, c=-1,\) and \(d=3\) $$2 a-(c+a)^{2}$$

Step-by-Step Solution

Answer

The evaluated result of the variable expression \(2 a-(c+a)^{2}\) for \(a=-2, c=-1\) is \(-13\).

Step 1: Substitute values of variables

Substitute \(a=-2\) and \(c=-1\) into the expression, we have, \(2(-2)-((-1)+(-2))^{2}\)

Step 2: Simplify inside the parentheses

Before doing any other calculations, we must simplify inside the parentheses, so we have, \(-4-(-3)^{2}\)

Step 3: Exponent Calculation

According to operator precedence, we should next calculate an exponent, it gives: \(-4-9\)

Step 4: Final Simplification

Finally, perform the subtraction to get the answer, which is \(-13\).

Key Concepts

Substitution in Variable Expressions

Substitution is a fundamental concept in algebra which involves replacing variables with their respective values. It's like replacing a placeholder in a sentence with an actual word that gives the sentence meaning. By substituting a variable with a given number, the abstract expression becomes a concrete numerical expression that you can calculate.

For instance, let's say we have an expression involving the variables \(a, b, c,\) and \(d\), and we are given the values \(a=-2, b=4, c=-1,\) and \(d=3\). Substituting these values into the original expression \(2a - (c + a)^2\) gives us a new expression \(2(-2) - ((-1) + (-2))^2\). This is the first critical step in evaluating the variable expression, as it sets the stage for subsequent calculation steps.

It's essential to substitute correctly to avoid any errors in calculation. Each variable must be replaced with its given value, and the substituted value must be contained within parentheses to maintain proper mathematical operations, especially if the substituted value is negative. This prevents sign errors which are common when dealing with negative numbers.

Operator Precedence

Understanding operator precedence, often referred to as the order of operations, is crucial when evaluating mathematical expressions. It dictates which calculations should be performed first in an expression with multiple operations. An easy way to remember the order is the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In our example, after substituting the values into the expression \(2(-2) - ((-1) + (-2))^2\), we need to follow the correct order of operations. The parentheses come first, so we simplify anything within them, leading to the next step which involves the exponents. Only after these steps, we can continue with multiplication, division, addition, or subtraction. By understanding and correctly applying the operator precedence, we can avoid common mistakes that lead to incorrect results.

Exponent Calculation

Calculating exponents is an integral part of evaluating mathematical expressions. An exponent tells you how many times to multiply the base by itself. For example, \(3^2\) means to multiply 3 by itself, resulting in 9.

In the case of the example expression \(2(-2) - ((-1) + (-2))^2\), after simplifying what's inside the parentheses we get \(2(-2) - (-3)^2\). The next step is to calculate the exponent \((-3)^2\). It’s important to recognize that a negative number squared (\(n^2\) where \(n\) is negative) will always result in a positive number. So, in our example, \( (-3)^2 = (-3) * (-3) = 9\) is the exponent calculation. Understanding how to correctly calculate exponents, especially with negative bases, is essential to simplify variable expressions accurately.