Problem 75 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ \left(0.03 f^{2}+0.25 f+0.91\right)-\left(0.17 f^{2}-1.18\right) $$

Step-by-Step Solution

Answer

The simplified expression is \( -0.14f^{2} + 0.25f + 2.09 \).

Step 1: Distribute the Negative Sign

When subtracting two expressions, distribute the negative sign to each term in the second expression: \(-(0.17f^{2} - 1.18) = -0.17f^{2} + 1.18\). This gives us: \(0.03f^{2} + 0.25f + 0.91 - 0.17f^{2} + 1.18\).

Step 2: Combine Like Terms

Group and combine like terms: - Combine \(f^{2}\) terms: \(0.03f^{2} - 0.17f^{2} = -0.14f^{2}\).- Combine constant terms: \(0.91 + 1.18 = 2.09\).This results in: \(-0.14f^{2} + 0.25f + 2.09\).

Step 3: Final Expression

After combining like terms, the simplified expression is: \( -0.14f^{2} + 0.25f + 2.09 \).

Key Concepts

Distributing Negative Sign

In mathematics, distributing the negative sign is a crucial step when dealing with subtraction between two expressions. Imagine you have a negative sign outside a parenthesis covering multiple terms within. The negative sign applies to each term inside the parenthesis.

This means for every term in the second expression, you change the sign:

• A positive term becomes negative.
• A negative term becomes positive.

Let's take the example:\(- (0.17f^{2} - 1.18)\). Here, you distribute the negative sign to each term:

• \(-0.17f^{2}\)
• \(+1.18\)

It's all about flipping the signs inside the parenthesis to simplify the expression later. By practicing this, you will become more comfortable handling expressions with parentheses.

Combining Like Terms

Combining like terms is an essential part of simplifying polynomial expressions. Similar terms, often called "like terms," are terms that have the same variables raised to the same power. This means they are similar enough to be added or subtracted together.

Let's work through an example with some terms:

• Terms containing \(f^{2}\): \(0.03f^{2}\) and \(-0.17f^{2}\) can be combined to form: \(-0.14f^{2}\).
• Constant terms like \(0.91\) and \(1.18\), which don't have a variable, can be added together: \(0.91 + 1.18 = 2.09\).

Combining like terms involves straightforward addition or subtraction. Look for terms that have the same variable and power. Simplifying these terms gives an expression that is easier to understand and work with.

Simplifying Expressions

Simplifying expressions makes them easier to evaluate and understand. Once you have distributed the negative sign and combined like terms, the next step is to simplify the resulting expression.

Simplification typically involves:

• Ensuring there are no parentheses.
• Combining all like terms as explained earlier.

For the expression resulting from our earlier steps, you reach a simplified form: \(-0.14f^{2} + 0.25f + 2.09\).

This single expression is neat and concise, representing all operations performed. Properly simplifying not only makes an expression easier to work with in further calculations but also improves overall mathematical readability.