Problem 35 - College Algebra

Question

Add or subtract as indicated. $$ \frac{x^{2}-2 x}{x^{2}+3 x}+\frac{x^{2}+x}{x^{2}+3 x} $$

Step-by-Step Solution

Answer

The result of the addition is \(\frac{2x² - x}{x² +3x}\).

Step 1: Identify the Common Denominator

The common denominator of the two fractions is \(x^{2}+3x\). It is the same in both fractions, so there's no need to find the lowest common denominator.

Step 2: Add or Subtract the Numerators

When the fractions have the same denominators, add or subtract the numerators based on the operation indicated. Here, we need to add the numerators. So, the equation becomes: \[(x² -2x) + (x²+x)\] Simplifying the above equation gives \(2x² -x\).

Step 3: Write the Sum or Difference over the Common Denominator

Now, write the sum or difference of the numerators over the common denominator. So, our final expression becomes \[\frac{2x² - x}{x² +3x}\]

Key Concepts

Common Denominator

In the world of fractions, a common denominator is like finding a shared stage for two performers. Both fractions need to "stand" on the same value to be properly added or subtracted.

In this exercise, the two fractions already have a common denominator: \(x^2 + 3x\). This makes our job much easier since there's no need to manipulate the denominators to make them the same.

When fractions share the same denominator, it's all about working with the numerators, leaving the shared denominator unchanged. Remember: no extra work needed if the denominators already match!

Numerical Simplification

Numerical simplification might sound complex, but it's simply the act of combining or rewriting numerical expressions in a simpler form. Once you have the same denominator, focus shifts to the numerators.

In our case, the expression states: \((x^2-2x) + (x^2+x)\). We need to combine these terms by collecting like terms (those that have the same variable and exponent).

By adding the coefficients of \( x^2 \) and \( x \), the result is \(2x^2-x\). Each step reduces the expression into something more straightforward, making the math easier to digest.

Simplification is a helpful tool to keep expressions neat and solutions manageable.

Algebraic Expressions

Algebraic expressions are like sentences in the language of algebra. They often contain numbers, variables, and operators (like addition or subtraction).

When working with these expressions, it's important to understand how to manipulate them correctly. In our fraction, the algebraic expressions are found in both the numerators and the denominators.

To tackle these, recognize pattern: terms with "\(x^2\)," terms with "\(x\)," and constant numbers.

Managing such terms allows you to add, subtract, and even factor the expressions if needed. They are the building blocks for solving not only this problem but also many others in algebra.

Fraction Addition

Adding fractions isn't too different from adding numbers: it's just a bit more elaborate due to denominators and numerators. With a common denominator in place, focus shifts to the top part—the numerators.

For this exercise, we start by simply adding the numerators \((x^2-2x) + (x^2+x)\). Then, carry out the addition keeping like terms together, bringing us to \(2x^2-x\).

Once added, place this new numerator back over the common denominator to form the final answer: \( \frac{2x^2 - x}{x^2 + 3x}\). This step-by-step adds precision to our fraction addition skills and reinforces the logical flow from problem to solution.