Problem 24 - Introductory Algebra for College Students

Question

add or subtract as indicated. Simplify the result, if possible. $$\begin{aligned} &\frac{x^{2}+9 x}{4 x^{2}-11 x-3}+\frac{3 x-5 x^{2}}{4 x^{2}-11 x-3}\\\ &x^{2}-4 x-4 x-4 \end{aligned}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-4x^{2} + 12x}{4x^{2}-11x-3} + x^{2}-8x-4 \)

Step 1: Addition of Fractions

When the denominators in the two fractions are the same, the fractions can be combined by adding the numerators together. So the first part of the expression becomes: \( \frac{x^{2}+9x+3x-5x^{2}}{4x^{2}-11x-3} \)

Step 2: Simplification

Simplify the expression in the numerator by combining the like terms. This gives: \( \frac{-4x^{2} + 12x}{4x^{2}-11x-3} \). Now, simplify the second polynomial \(x^{2}-4x-4x-4\) by combining like terms to get \(x^{2}-8x-4\).

Step 3: Final Solution

Now, write the final solution combining both simplified parts. The final solution will be \( \frac{-4x^{2} + 12x}{4x^{2}-11x-3} + x^{2}-8x-4 \).

Key Concepts

Understanding the Addition of Fractions

Adding fractions might seem tricky at first, but it's all about having the same denominator. Here, both fractions share a common denominator of \(4x^{2}-11x-3\). When denominators are the same, you only need to add the numerators together:

• The expression given is: \(\frac{x^{2}+9x}{4x^{2}-11x-3} + \frac{3x - 5x^{2}}{4x^{2}-11x-3}\).
• Since they have the same denominator, we combine the numerators: \(x^{2} + 9x + 3x - 5x^{2}\).

This results in a new fraction: \(\frac{x^{2}+9x+3x-5x^{2}}{4x^{2}-11x-3}\). Adding fractions becomes a piece of cake when they share denominators.Knowing how to correctly add numerators is crucial before diving into further simplification.

Simplifying Algebraic Expressions

Simplifying expressions is like decluttering a messy workspace to only keep what you need. The goal is to combine like terms, making your expression as neat as possible. Here’s how we simplify:

• Look at the combined numerators: \(x^{2} + 9x + 3x - 5x^{2}\).
• Rearrange the terms to group like ones: \(-4x^{2} + 12x\).

Notice how we bundled all \(x^{2}\) terms and all \(x\) terms together. From here, the expression simplifies to: \(\frac{-4x^{2} + 12x}{4x^{2}-11x-3}\).Simplifying is important to make calculations easier to handle in future steps.

Combining Like Terms

Combining like terms is crucial in algebra to simplify expressions. It involves merging terms in an expression that have the same variable to reduce and clarify the expression:

• We initially have \(x^{2} - 4x - 4x - 4\).
• Combine the \(x\) terms: \(-4x\) and \(-4x\) yield \(-8x\).

The expression simplifies to \(x^{2} - 8x - 4\).This process reduces complexity and provides clarity, setting the stage for further operations or solving equations. It's like finding the best path through the forest by clearing away the underbrush.