Problem 29 - Algebra 2

Question

Evaluate each expression if \(a=\frac{2}{5}, b=-3, c=0.5,\) and \(d=6\). \((a-c)^{2}-2 b d\)

Step-by-Step Solution

Answer

The value of the expression is 36.01.

Step 1: Substitute the Values

First, we replace the variables in the expression \((a-c)^{2}-2bd\) with the given values. We have \(a = \frac{2}{5}\), \(b = -3\), \(c = 0.5\), and \(d = 6\). Thus, substitute these values into the expression to get \[\left(\frac{2}{5} - 0.5\right)^2 - 2(-3)(6).\]

Step 2: Evaluate Inside the Parentheses

Now, calculate the expression inside the parentheses: \(\frac{2}{5} - 0.5 = \frac{2}{5} - \frac{1}{2}\).Convert \(0.5\) to a fraction with the same denominator as \(\frac{2}{5}\), which is \(\frac{5}{10}\), then convert it to \(\frac{2.5}{5}\). Hence,\[\frac{2}{5} - \frac{2.5}{5} = \frac{-0.5}{5} = \frac{-1}{10}.\]

Step 3: Square the Result

Next, square the result from Step 2: \[\left(\frac{-1}{10}\right)^2 = \frac{1}{100}.\]

Step 4: Evaluate the Product

Calculate the second part of the expression:\(-2bd\) where \(b = -3\) and \(d = 6\)\(-2(-3)(6) = 36\).

Step 5: Combine the Results

Now, combine the results from Step 3 and Step 4 to find the final value of the expression. Adding \(\frac{1}{100}\) and \(36\) gives: \[\frac{1}{100} + 36 = 36 + 0.01 = 36.01.\]

Key Concepts

Substitution

Substitution in algebra is about replacing variables with their actual values. Imagine you have an expression featuring various letters that stand for numbers. Your task is to swap these letters for the numbers they represent. For example, in the expression \((a-c)^{2}-2bd\), we're given that \(a = \frac{2}{5}\), \(b = -3\), \(c = 0.5\), and \(d = 6\). To substitute, simply replace each variable in the expression with the corresponding number.
This gives us:

• Replace \(a\) with \(\frac{2}{5}\)
• Replace \(c\) with \(0.5\)
• Do the same for \(b\) and \(d\)

Ultimately, the substituted expression is \(\left(\frac{2}{5} - 0.5\right)^2 - 2(-3)(6)\).
Substitution helps simplify expressions and prepare them for further calculations.

Arithmetic Operations

Arithmetic operations are the basic building blocks of mathematics. They include addition, subtraction, multiplication, and division. When evaluating expressions, you will use these operations to simplify and solve them.
In our example, within the expression \(\left(\frac{2}{5} - 0.5\right)^2 - 2(-3)(6)\), arithmetic operations help determine the value of the expression:

• Subtract \(0.5\) from \(\frac{2}{5}\)
• Multiply \(-3\) by \(6\)
• Simplify these to arrive at some results you can further work with

Understanding and applying the right operation is key to solving problems accurately.

Order of Operations

The order of operations is critical when dealing with mathematical expressions. It guides us on the correct sequence to tackle the operations available in an expression to arrive at the right result. The acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is often used to remember the order.
In the expression we evaluated, the order of operations is applied as follows:

• First, solve operations inside the parentheses: \(\frac{2}{5} - 0.5\)
• Next, deal with the exponent: square the result obtained from the parentheses
• Then, perform multiplication and division from left to right: calculate \(-2(-3)(6)\)
• Lastly, carry out any addition or subtraction left: adding \(\frac{1}{100}\) and \(36\)

Following this structured approach ensures every part of the expression is handled in the proper order, avoiding mistakes.

Fraction Arithmetic

Working with fractions combines some basic arithmetic with the rules peculiar to fractions. When subtracting fractions, you need to have a common denominator to perform the operation efficiently. Let's see this in action with our example.
In the expression part \(\frac{2}{5} - 0.5\), we need to convert \(0.5\) to a fraction with a denominator of 5. It becomes \(\frac{2.5}{5}\). This simplifies the subtraction:

• Perform the subtraction: \(\frac{2}{5} - \frac{2.5}{5} = \frac{-0.5}{5}\)
• Then it further simplifies to \(\frac{-1}{10}\)

Handling fractions accurately is crucial, as it often affects the result significantly. Practice helps make these operations straightforward.