Add the following and simplify the sum if possible. x 2 − 4 x 2 − x − 6 + x + 2 x 2 − x − 6 \frac{x^2-4}{x^2-x-6}+\frac{x+2}{x^2-x-6}

This problem asks us to add the following and simplify the sum if possible. So we have x squared minus four over x squared minus x minus six, and then we have all of this plus x plus two over x squared minus x minus six. Now what we need to do for rational expressions when we're adding or subtracting is look for common denominators. Now I can see we have the same quadratic, the same denominator, so that means we can just combine the numerators. So we have x squared minus four for the first rational expression, then we have x plus two for the second, then all of this is gonna be over the common denominator, x squared minus x minus six. Now notice in the numerator, we don't have any values, we don't have any numbers that are being multiplied on the outside of these parentheses, nor do we have any negative signs or subtractions. So this whole thing is just gonna be x squared minus four plus x plus two, and then the denominator is going to remain x squared minus x minus six. Now what can I do from here? Well what I can do is combine things and simplify in the numerator. Now I notice we have an x squared, that's one term, and then we have an x, that's another term. Then I notice we have a negative four and positive two. Negative four plus two will give us negative two. So we're gonna end up with x squared plus x, and then the negative two will just be a minus two that we can write like that. And we're gonna have x squared minus x minus six. Now it may be tempting to think we can stop here, but we need to simplify this as far as it will go. But how can we simplify farther from this point? Well, notice I end up with two quadratics in the numerator and the denominator. Now I'm gonna write these separately. So we have x squared plus x minus two that's happening in the numerator, and then we have x squared minus x minus six in the denominator. And what I need to do is factor both of these. So I need to think about a pair of numbers for the numerator, the the top quadratic, that is going to multiply to give me negative two, and it's going to add to give me positive one. There's an implicit one that we have right here. So what's going to multiply to give me negative two and add to give me positive one? Well, if we think about through all these pairs of numbers, that's going to be two and negative one. Right? Because two times negative one gives us negative two, and then two plus negative one gives us positive one. So we've gone ahead and found this pair of numbers. Now what I'm going to do is also find the pair of numbers for the other quadratic, x squared minus x minus six. I need to find a pair of numbers that will multiply to give me negative six, and they'll add to give me negative one. What pair of numbers is that going to be? Well if I think about it, that's going to be two and negative three. And we can just check two times negative three, that's negative six. Two plus negative three, that's negative one. So we have found this pair of numbers. So now that we found each of these pairs, we can continue on by factoring what we have. So we're going to keep the, we're going to have this fraction bar here and then we're going to have in the numerator, we're going to have the factored version of this. We found the pair of numbers was two and negative one, so it's going to be x plus two, but positive two, and x minus one, the negative one. Now in the denominator, we got positive two and negative three. So we're going to have x plus two and then x minus three, that's what happens when we factor. And notice here that the x plus two cancels entirely. So all we end up with is x minus one over x minus three. And this cannot simplify down any farther, so this right here would be our final answer. So that's how you can solve these problems with rational expressions. As you can see, it's oftentimes the simplifying step that takes a long time, and you will often have to do multiple steps to get to your fully simplified answer. So make sure whatever answer you end up with, it's something that cannot be factored or broken down any farther and something that will not simplify any farther. So So I hope you found this video helpful, and let's move on.

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1. Review of Real Numbers2h 39m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
40m

2. Linear Equations and Inequalities3h 38m

The Addition and Subtraction Properties of Equality
41m

The Multiplication and Division Properties of Equality
30m

Solving Linear Equations
1h 14m

Linear Inequalities in One Variable
1h 11m

3. Solving Word Problems2h 43m

Introduction to Problem Solving
37m

Formulas
21m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphing Linear Equations in Two Variables3h 17m

The Rectangular Coordinate System
44m

Graph Linear Equations in Two Variables
24m

Graph Linear Equations Using Intercepts
23m

Slope of a Line
44m

Slope-Intercept Form
38m

Point Slope Form
22m

5. Systems of Linear Equations1h 43m

Solving Systems of Linear Equations by Graphing
41m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

6. Exponents and Polynomials3h 25m

The Product Rule
10m

The Quotient Rule
13m

Intro to the Power Rules
18m

The Power of a Quotient Rule
18m

Negative Exponents
28m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

Adding and Subtracting Polynomials
20m

Multiplying Polynomials
33m

Special Products
34m

7. Factoring2h 42m

Factoring by Greatest Common Factor & Grouping
38m

Factoring Trinomials of the Form x² + bx + c
11m

Factoring Trinomials of the Form ax² + bx + c
37m

Factoring Special Products
33m

Quadratic Equations & Applications
41m

8. Rational Expressions and Equations3h 13m

Simplifying Rational Expressions
39m

Multiplying and Dividing Rational Expressions
25m

Adding and Subtracting Rational Expressions with Common Denominators
19m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

9. Inequalities and Absolute Value2h 52m

Set Operations and Compound Inequalities
42m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

Linear Inequalities in Two Variables
28m

Systems of Linear Inequalities
23m

10. Relations and Functions2h 9m

Introduction to Relations and Functions
57m

Function Notation
15m

Graphing Common Functions
5m

Composition of Functions
24m

Direct & Inverse Variation
27m

11. Roots, Radicals, and Complex Numbers3h 57m

Introduction to Radical Expressions
47m

Simplifying Radical Expressions
47m

Adding and Subtracting Radicals
19m

Multiplying, Dividing, and Rationalizing Radicals
53m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

12. Quadratic Equations and Functions3h 1m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
40m

Graphing Quadratic Equations
1h 28m

13. Inverse, Exponential, & Logarithmic Functions2h 31m

Introduction to Inverse Functions
25m

Exponential Functions
35m

Introduction to Logarithmic Functions
33m

Properties of Logarithms
55m

14. Conic Sections & Systems of Nonlinear Equations2h 24m

Graphing Circles
32m

Ellipses
35m

Parabolas
35m

Hyperbolas
41m

15. Sequences, Series, and the Binomial Theorem1h 45m

Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
26m

Factorials
11m

8. Rational Expressions and Equations

Adding and Subtracting Rational Expressions with Common Denominators

Beginning & Intermediate Algebra

8. Rational Expressions and Equations

Adding and Subtracting Rational Expressions with Common Denominators

Problema precedenteProssimo problema

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Add the following and simplify the sum if possible.
$\frac{x^{2} - 4}{x^{2} - x - 6} + \frac{x + 2}{x^{2} - x - 6}$

$\frac{x + 2}{x - 3}$

$\frac{x + 2}{x + 3}$

$\frac{x - 1}{x - 3}$

$\frac{x - 1}{x + 2}$

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Problema precedenteProssimo problema

Guida verificata passo dopo passo

Identify that both fractions have the same denominator: $x^2 - x - 6$. This means you can add the numerators directly over the common denominator.

Write the sum as a single fraction: $\frac{x^2 - 4}{x^2 - x - 6} + \frac{x + 2}{x^2 - x - 6} = \frac{(x^2 - 4) + (x + 2)}{x^2 - x - 6}$.

Simplify the numerator by combining like terms: $x^2 - 4 + x + 2 = x^2 + x - 2$.

Factor both the numerator and the denominator to see if any common factors can be canceled. For the numerator, factor $x^2 + x - 2$. For the denominator, factor $x^2 - x - 6$.

After factoring, cancel any common factors between numerator and denominator to simplify the fraction to its lowest terms.