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Chapter 0: Problem 35

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

Short Answer

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

Step by step solution

01

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.
02

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: \[(x虏 -2x) + (x虏+x)\] Simplifying the above equation gives \(2x虏 -x\).
03

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 \[\frac{2x虏 - x}{x虏 +3x}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥攖he 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.

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Most popular questions from this chapter

Use the product rule to simplify the expressions in Exercises \(7-16 .\) In Exercises \(11-16,\) assume that variables represent nonnegative real numbers. $$\sqrt{10 x} \cdot \sqrt{8 x}$$

In Exercises \(11-16,\) factor by grouping. $$3 x^{3}-2 x^{2}-6 x+4$$

In Exercises \(11-16,\) factor by grouping. $$x^{3}-3 x^{2}+4 x-12$$

Use the quotient rule to simplify the expressions in Exercises \(17-26 .\) Assume that \(x>0\) $$\frac{\sqrt{24 x^{4}}}{\sqrt{3 x}}$$

Use the quotient rule to simplify the expressions in Exercises \(17-26 .\) Assume that \(x>0\) $$\frac{\sqrt{48 x^{3}}}{\sqrt{3 x}}$$
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