Add the following expressions and simplify if possible: x 2 x 2 − 4 + 2 x 4 − x 2 \frac{x^2}{x^2-4}+\frac{2x}{4-x^2}

this problem we are asked to add the following expressions and simplify if possible. Now I can see I'm adding two rational expressions. Now what I'm going to do is see how I can factor the numerators and denominators completely of each. This is going to make adding them together a bit more simple, since we're ultimately going to find the least common denominator, or LCD, to solve this problem. Now in the numerator, I can see we have this x squared. And then in the denominator, we have x squared minus four. Four is the same thing as two squared. And notice how I get us to have a difference of two squares right about here. So what I can do is take this whole denominator, x squared of minus two squared, and I can factor that to be x plus two times x minus two. This is the rule for when we have a difference of two squares. So x squared minus four factors to become x plus two times x minus two. Now what I'm going to do is add this to this expression we have on the right. So that's going to be two x over, and then we're going to have four minus x squared. Now what I'm going to do is take this x squared and make it negative. So it's the same thing as four plus negative x squared. But what I can do is flip around the order since we're now adding these, so I can bring this negative x squared to the front, giving us negative x squared plus four. Now what I can do from here is factor out a negative one from this denominator. So we're going to keep everything the same here, except I'm going to factor a negative one out of the denominator. So if I factor a negative one out of the denominator, well that's going to leave me with a negative one on the outside of the parenthesis, or just a negative sign, and then on the inside we'll change the signs of each of these, so that's going to give me x squared, and then we'll have minus four. Now notice x squared minus four is the same thing as x squared minus two squared. This is a difference of squares. So I can go through this process of factoring again. So we'll keep everything the same, but I'm going to use this difference of squares that I've been using in these problems. So this x squared minus two, that's going to become x plus two times x minus two. That's x squared minus two squared. Now we have close common denominators. The main difference that I'm seeing is that we don't have a negative sign over here. Now I could do this a couple different ways. I could multiply the top and bottom by a negative one in this left expression, or what I could do is I could simply bring this negative sign to the front of this expression on the right. And that's what I'm going to do, because a negative sign would just make this whole thing subtracted. So what we're ultimately gonna end up with on top is x squared minus two x, and then what we'll end up with on bottom is x plus two times x minus two. Now what I'm gonna do is factor out this x from each of these terms. We'll have x times x minus two, and then the denominator will have x plus two times x minus two. And I can go ahead and cancel those x minus two's. So all this is going to leave me with is x in the numerator and x plus two in the denominator. And that is how we can solve this problem. Hope you found this video helpful.

Add the following expressions and simplify if possible:x2x2−4+2x4−x2\frac{$$x^2$$}{$$x^2-4$$}+\frac{2x}{$$4-x^2$$}

x2(x−2)(x+2)\frac{$$x^2$$}{\left(x-2\right)\left(x+2\right)}(x−2)(x+2)x2

Table of contents

Skip topic navigation

1. Review of Real Numbers2h 43m

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
44m

2. Linear Equations and Inequalities5h 35m

Solving Linear Equations
2h 23m

Linear Inequalities in One Variable
46m

Set Operations and Compound Inequalities
1h 7m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

3. Solving Word Problems2h 46m

Introduction to Problem Solving
37m

Formulas
25m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphs and Functions5h 12m

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

Linear Inequalities in Two Variables
28m

Introduction to Relations and Functions
53m

Function Notation
15m

Composition of Functions
17m

5. Systems of Linear Equations1h 53m

Solving Systems of Linear Equations by Graphing
29m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

Systems of Linear Inequalities
23m

6. Exponents, Polynomials, and Polynomial Functions3h 17m

The Product Rule
9m

The Quotient Rule
10m

Intro to the Power Rules
17m

The Power of a Quotient Rule
13m

Negative Exponents
24m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

Adding and Subtracting Polynomials
20m

Multiplying Polynomials
38m

Special Products
34m

7. Factoring2h 49m

Factoring by Greatest Common Factor & Grouping
37m

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

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

Factoring Special Products
41m

Quadratic Equations & Applications
41m

8. Rational Expressions and Functions3h 44m

Simplifying Rational Expressions
42m

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

Direct & Inverse Variation
27m

9. 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

10. Quadratic Equations and Functions3h 1m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
40m

Graphing Quadratic Equations
1h 28m

11. Inverse, Exponential, & Logarithmic Functions2h 30m

Introduction to Inverse Functions
25m

Exponential Functions
35m

Introduction to Logarithmic Functions
33m

Properties of Logarithms
54m

12. Conic Sections & Systems of Nonlinear Equations2h 24m

Graphing Circles
32m

Ellipses
35m

Parabolas
35m

Hyperbolas
41m

13. Sequences, Series, and the Binomial Theorem1h 51m

Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
29m

Factorials
13m

8. Rational Expressions and Functions

Adding and Subtracting Rational Expressions with Different Denominators

Intermediate Algebra

8. Rational Expressions and Functions

Adding and Subtracting Rational Expressions with Different Denominators

Previous problemNext problem

Struggling with Intermediate Algebra?

Join thousands of students who trust us to help them ace their exams!

Multiple Choice

Add the following expressions and simplify if possible:
$\frac{x^{2}}{x^{2} - 4} + \frac{2 x}{4 - x^{2}}$

$\frac{x^2}{\left(x-2\right)\left(x+2\right)}$

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

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

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

0 Comments

Previous problemNext problem

Verified step by step guidance

Identify the given expressions: $\frac{x^2}{x^2 - 4} + \frac{2x}{4 - x^2}$. Notice that the denominators are very similar but not exactly the same.

Factor the denominators to see their relationship: $x^2 - 4$ factors as $(x - 2)(x + 2)$, and $4 - x^2$ can be rewritten as $-(x^2 - 4)$, so it factors as $-(x - 2)(x + 2)$.

Rewrite the second fraction using the factored form and the negative sign: $\frac{2x}{4 - x^2} = \frac{2x}{-(x - 2)(x + 2)} = -\frac{2x}{(x - 2)(x + 2)}$.

Now express both fractions with the common denominator $(x - 2)(x + 2)$: $\frac{x^2}{(x - 2)(x + 2)} - \frac{2x}{(x - 2)(x + 2)}$.

Combine the numerators over the common denominator: $\frac{x^2 - 2x}{(x - 2)(x + 2)}$. Then factor the numerator if possible to simplify the expression further.

8:03m

Watch next

Master Adding and Subtracting Rational Expressions with Unlike Denominators Example 1 with a bite sized video explanation from Patrick Ford

Struggling with Intermediate Algebra?

Add the following expressions and simplify if possible:x2x2−4+2x4−x2\(\frac{x^2}{x^2-4}\)+\(\frac{2x}{4-x^2}\)

Watch next

Add the following expressions and simplify if possible:
$\frac{x^{2}}{x^{2} - 4} + \frac{2 x}{4 - x^{2}}$